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Auteur : Grafiati

Publié le 11 mars 2023

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1

Kranysˇ, M. « Causal Theories of Evolution and Wave Propagation in Mathematical Physics ». Applied Mechanics Reviews 42, n^o 11 (1 novembre 1989) : 305–22. http://dx.doi.org/10.1115/1.3152415.

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There are still many phenomena, especially in continuum physics, that are described by means of parabolic partial differential equations whose solution are not compatible with the causality principle. Compatibility with this principle is required also by the theory of relativity. A general form of hyperbolic operators for the most frequently occurring linear governing equations in mathematical physics is written down. It is then easy to convert any given parabolic equation to the hyperbolic form without necessarily entering into the cause of the inadequacy of the governing equation. The method is verified on the well-known example of Timoshenko’s correction of the Bernoulli–Euler–Rayleigh beam equation for flexural motion. The “Love–Rayleigh” fourth-order differential equations for the longitudinal and torsional wave propagation in the rod is generalized with this method. The hyperbolic version (not to mention others) of the linear Korteweg–de Vries equation and of the “telegraph” equation governing electromagnetic wave propagation through relaxing material are given. Lagrangians of all the equations studied are listed. For all the reasons given we believe the hyperbolic governing equations to be physically and mathematically more realistic and adequate.

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2

Ashyralyev, Allaberen, Yasar Sozen et Fatih Hezenci. « A note on evolution equation on manifold ». Filomat 35, n^o 15 (2021) : 5031–43. http://dx.doi.org/10.2298/fil2115031a.

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In the present paper, considering the differential equations on smooth closed manifolds, we investigate and establish the well-posedness of boundary value problems nonlocal type for parabolic equations and also hyperbolic equations in H?lder spaces. Furthermore, in various H?lder norms we establish new coercivity estimates for the solutions of such type parabolic boundary value problems on manifolds and hyperbolic boundary value problems on manifolds as well.

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3

Avalishvili, Gia, et Mariam Avalishvili. « On nonclassical problems for first-order evolution equations ». gmj 18, n^o 3 (14 juillet 2011) : 441–63. http://dx.doi.org/10.1515/gmj.2011.0028.

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Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

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4

Colli, Pierluigi, Gianni Gilardi et Jürgen Sprekels. « Constrained Evolution for a Quasilinear Parabolic Equation ». Journal of Optimization Theory and Applications 170, n^o 3 (6 juillet 2016) : 713–34. http://dx.doi.org/10.1007/s10957-016-0970-6.

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Vidossich, Giovanni. « Solving Becker’s Problem on Periodic Solutions of Parabolic Evolution Equations ». Advanced Nonlinear Studies 18, n^o 2 (1 avril 2018) : 195–215. http://dx.doi.org/10.1515/ans-2017-6047.

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Abstract We present existence and multiplicity theorems for periodic mild solutions to parabolic evolution equations. Their peculiarity is a link with the spectrum of the generator of the semigroup rather than with the spectrum of the linearized periodic BVP for the evolution equation. They provide a positive solution to the open problem risen by Becker [3], they extend some results of Castro and Lazer [5] from scalar to systems of parabolic equations, and they are new even for finite-dimensional ODEs.

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6

LIN, CHIN-YUAN. « SOME NON-DISSIPATIVITY CONDITION FOR EVOLUTION EQUATIONS ». International Journal of Mathematics 24, n^o 02 (février 2013) : 1350002. http://dx.doi.org/10.1142/s0129167x1350002x.

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Of concern is the nonlinear evolution equation [Formula: see text] in a real Banach space X, where the nonlinear, time-dependent, multi-valued operator [Formula: see text] has a time-dependent domain D(A(t)). It will be shown that, under some non-dissipativity condition, the equation has a strong solution. Illustrations are given of solving quasi-linear partial differential equations of parabolic type.

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7

Karachalios, Nikos, Nikos Stavrakakis et Pavlos Xanthopoulos. « Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation ». Abstract and Applied Analysis 2003, n^o 9 (2003) : 521–38. http://dx.doi.org/10.1155/s1085337503210022.

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We consider a nonlinear parabolic equation involving nonmonotone diffusion. Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global attractor for the corresponding dynamical system.

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8

Motsa, S. S., V. M. Magagula et P. Sibanda. « A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations ». Scientific World Journal 2014 (2014) : 1–13. http://dx.doi.org/10.1155/2014/581987.

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This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

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9

Su, Ning, et Li Zhang. « Existence for Nonlinear Evolution Equations and Application to Degenerate Parabolic Equation ». Journal of Applied Mathematics 2014 (2014) : 1–8. http://dx.doi.org/10.1155/2014/567241.

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We consider an abstract Cauchy problem for a doubly nonlinear evolution equation of the formd/dt𝒜u+ℬu∋ftinV′,t∈0, T, whereVis a real reflexive Banach space,𝒜andℬare maximal monotone operators (possibly multivalued) fromVto its dualV′. In view of some practical applications, we assume that𝒜andℬare subdifferentials. By using the back difference approximation, existence is established, and our proof relies on the continuity of𝒜and the coerciveness ofℬ. As an application, we give the existence for a nonlinear degenerate parabolic equation.

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10

Bulíček, Miroslav, Piotr Gwiazda, Endre Süli et Agnieszka Świerczewska-Gwiazda. « Analysis of a viscosity model for concentrated polymers ». Mathematical Models and Methods in Applied Sciences 26, n^o 08 (7 juin 2016) : 1599–648. http://dx.doi.org/10.1142/s0218202516500391.

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The paper is concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier–Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic–hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient, appearing in the balance of linear momentum equation in the Navier–Stokes system, includes dependence on the shear rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We prove the existence of global-in-time, large-data weak solutions under fairly general hypotheses.

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11

Kudrych, Yuliia, et Mariia Savchenko. « Removable isolated singularities for anisotropic of anisotropic evolution p-Laplacian equation ». Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 35 (28 janvier 2022) : 137–51. http://dx.doi.org/10.37069/1683-4720-2021-35-10.

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In this paper we study nonnegative weak solutions of a quasilinear parabolic equations in a divergent form, one of the model cases of which is anisotropic evolution p-Laplacian equation. New pointwise estimates near an isolated singularity and sufficient condition of the removability of isolated singularity were obtained.

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12

Wang, R. N., et Y. Zhou. « Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations ». Abstract and Applied Analysis 2014 (2014) : 1–11. http://dx.doi.org/10.1155/2014/263690.

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This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the formu′(t)=A(t)u(t)+f(t,u(t)),t∈R,u(t+T)=-u(t),t∈R, where(At)t∈R(possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach spaceX. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on(At)t∈R, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.

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13

D’Acunto, Berardino, Luigi Frunzo, Vincenzo Luongo et Maria Rosaria Mattei. « Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm ». Mathematics 7, n^o 9 (24 août 2019) : 781. http://dx.doi.org/10.3390/math7090781.

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A mathematical model able to simulate the physical, chemical and biological interactions prevailing in multispecies biofilms in the presence of a toxic heavy metal is presented. The free boundary value problem related to biofilm growth and evolution is governed by a nonlinear ordinary differential equation. The problem requires the integration of a system of nonlinear hyperbolic partial differential equations describing the biofilm components evolution, and a systems of semilinear parabolic partial differential equations accounting for substrates diffusion and reaction within the biofilm. In addition, a semilinear parabolic partial differential equation is introduced to describe heavy metal diffusion and sorption. The biosoption process modeling is completed by the definition and integration of other two systems of nonlinear hyperbolic partial differential equations describing the free and occupied binding sites evolution, respectively. Numerical simulations of the heterotrophic-autotrophic interaction occurring in biofilm reactors devoted to wastewater treatment are presented. The high biosorption ability of bacteria living in a mature biofilm is highlighted, as well as the toxicity effect of heavy metals on autotrophic bacteria, whose growth directly affects the nitrification performance of bioreactors.

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14

Short, M. « A parabolic linear evolution equation for cellular detonation instability ». Combustion Theory and Modelling 1, n^o 3 (mars 1997) : 313–46. http://dx.doi.org/10.1088/1364-7830/1/3/005.

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15

CHAVES, MANUELA, et VICTOR A. GALAKTIONOV. « Regional blow-up for a higher-order semilinear parabolic equation ». European Journal of Applied Mathematics 12, n^o 5 (octobre 2001) : 601–23. http://dx.doi.org/10.1017/s0956792501004685.

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We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation[formula here]with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution[formula here]of the complex Hamilton–Jacobi equation[formula here].

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16

Yang, He, et Yue Liang. « Positive Solutions for the Initial Value Problem of Fractional Evolution Equations ». Abstract and Applied Analysis 2013 (2013) : 1–7. http://dx.doi.org/10.1155/2013/428793.

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By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.

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17

Kurima, Shunsuke. « Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems ». ESAIM : Mathematical Modelling and Numerical Analysis 54, n^o 3 (22 avril 2020) : 977–1002. http://dx.doi.org/10.1051/m2an/2019086.

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This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter (see e.g., Grasselli and Pata [Adv. Math. Sci. Appl. 13 (2003) 443–459, Comm. Pure Appl. Anal. 3 (2004) 849–881], Grasselli et al. [Comm. Pure Appl. Anal. 5 (2006) 827–838], Wu et al. [Math. Models Methods Appl. Sci. 17 (2007) 125–153, J. Math. Anal. Appl. 329 (2007) 948–976]). On the other hand, a time discretization of an initial value problem for an abstract evolution equation has been studied (see e.g., Colli and Favini [Int. J. Math. Math. Sci. 19 (1996) 481–494]) and Schimperna [J. Differ. Equ. 164 (2000) 395–430] has established existence of solutions to an abstract problem applying to a nonlinear phase-field system of Caginalp type on a bounded domain by employing a time discretization scheme. In this paper we focus on a time discretization of a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. Moreover, we can establish an error estimate for the difference between continuous and discrete solutions.

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18

Mrozek, Marian, et Krzysztof P. Rybakowski. « Nontrivial full bounded solutions of time-periodic semilinear parabolic PDEs ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 117, n^o 3-4 (1991) : 305–15. http://dx.doi.org/10.1017/s0308210500024756.

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SynopsisConsider the semilinear evolution equation(P) u + Au = f(t,u)where A is a sectorial operator on a Banach space and f is ω-periodic in t. Using a time-discrete Conley index developed in a previous paper [6], we prove a few existence results on bounded solutions of (P) defined for all t ∊ R. More specific results are given for time-periodic scalar parabolic equations.

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19

Carl, Siegfried. « Existence of extremal periodic solutions for quasilinear parabolic equations ». Abstract and Applied Analysis 2, n^o 3-4 (1997) : 257–70. http://dx.doi.org/10.1155/s1085337597000389.

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In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.

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20

Agresti, Antonio, et Mark Veraar. « Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence* ». Nonlinearity 35, n^o 8 (13 juillet 2022) : 4100–4210. http://dx.doi.org/10.1088/1361-6544/abd613.

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Abstract In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L p -setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers’ equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

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21

MORALES-RODRIGO, CRISTIAN, et J. IGNACIO TELLO. « GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF A TUMOR ANGIOGENESIS MODEL WITH CHEMOTAXIS AND HAPTOTAXIS ». Mathematical Models and Methods in Applied Sciences 24, n^o 03 (29 décembre 2013) : 427–64. http://dx.doi.org/10.1142/s0218202513500553.

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We consider a system of differential equations modeling tumor angiogenesis. The system consists of three equations: two parabolic equations with chemotactic terms to model endothelial cells and tumor angiogenesis factors coupled to an ordinary differential equation which describes the evolution of the fibronectin concentration. We study global existence of solutions and, under extra assumption on the initial data of the fibronectin concentration we obtain that the homogeneous steady state is asymptotically stable.

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22

Selmani, Mohamed, et Lynda Selmani. « Analysis of a frictionless contact problem for elastic-viscoplastic materials ». Nonlinear Analysis : Modelling and Control 17, n^o 1 (25 janvier 2012) : 99–117. http://dx.doi.org/10.15388/na.17.1.14081.

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We consider a dynamic frictionless contact problem for elastic-viscoplastic materials with damage. The contact is modelled with normal compliance condition. The adhesion of the contact surfaces is considered and is modelled with a surface variable, the bonding field whose evolution is described by a first order differential equation. We derive variational formulation for the model and prove an existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed-point arguments.

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23

Papageorgiou, N. « Optimality conditions for systems driven by nonlinear evolution equations ». Mathematical Problems in Engineering 1, n^o 1 (1995) : 27–36. http://dx.doi.org/10.1155/s1024123x95000044.

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Using the Dubovitskii-Milyutin theory we derive necessary and sufficient conditions for optimality for a class of Lagrange optimal control problems monitored by a nonlinear evolution equation and involving initial and/or terminal constraints. An example of a parabolic control system is also included.

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24

Lo Giudice, A., G. Giammanco, D. Fransos et L. Preziosi. « Modeling sand slides by a mechanics-based degenerate parabolic equation ». Mathematics and Mechanics of Solids 24, n^o 8 (12 juin 2018) : 2558–75. http://dx.doi.org/10.1177/1081286518755230.

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Avalanching plays a crucial role in granular materials dynamics, in particular in the evolution of the shape of the leeward side of sand dunes. This paper presents a physically-based mathematical model capable of reproducing the kinematic evolution of the surface of sand piles and to obtain eventually the stationary configurations, in the presence of external sources as well. Simulation results with different boundary conditions and geometries are reported in order to show the high flexibility of the model. The model is also validated by means of comparison with the experimental results of different authors.

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25

Lorenzi, Alfredo, et Ioan I. Vrabie. « Identification of a Source Term in a Semilinear Evolution Delay Equation ». Annals of the Alexandru Ioan Cuza University - Mathematics 61, n^o 1 (1 janvier 2015) : 1–39. http://dx.doi.org/10.2478/aicu-2013-0003.

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Abstract An existence, uniqueness and continuous dependence on the data result for a source term identification problem in a semilinear functional delay differential equation in a general Banach space is established. As additional condition, it is assumed that the mean of the solution, with respect to a non-atomic Borel measure, is a preassigned element in the domain of the linear part of the right-hand side of the equation. Two applications to source identification, one in a parabolic functional delay equation and another one in a hyperbolic delay equation, are also discussed.

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26

Ben Alaya, Mohamed, et Benjamin Jourdain. « Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology ». Journal of Applied Probability 44, n^o 2 (juin 2007) : 528–46. http://dx.doi.org/10.1239/jap/1183667419.

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In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

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Ben Alaya, Mohamed, et Benjamin Jourdain. « Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology ». Journal of Applied Probability 44, n^o 02 (juin 2007) : 528–46. http://dx.doi.org/10.1017/s0021900200118005.

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In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

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Ben Alaya, Mohamed, et Benjamin Jourdain. « Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology ». Journal of Applied Probability 44, n^o 02 (juin 2007) : 528–46. http://dx.doi.org/10.1017/s0021900200003144.

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In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

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Liu, Changchun, Junchao Gao et Songzhe Lian. « Existence of Solutions for the Evolution -Laplacian Equation Not in Divergence Form ». Journal of Applied Mathematics 2012 (2012) : 1–21. http://dx.doi.org/10.1155/2012/835495.

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The existence of weak solutions is studied to the initial Dirichlet problem of the equation , with inf . We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

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Long, Nguyen Thanh, et Alain Pham Ngoc Dinh. « Approximation of a parabolic non-linear evolution equation backwards in time ». Inverse Problems 10, n^o 4 (1 août 1994) : 905–14. http://dx.doi.org/10.1088/0266-5611/10/4/010.

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Poon, Chi-Cheung, et Dong-Ho Tsai. « On a nonlinear parabolic equation arising from anisotropic plane curve evolution ». Journal of Differential Equations 258, n^o 7 (avril 2015) : 2375–407. http://dx.doi.org/10.1016/j.jde.2014.12.010.

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32

Coronel, Aníbal, Fernando Huancas, Alex Tello et Marko Rojas-Medar. « New Necessary Conditions for the Well-Posedness of Steady Bioconvective Flows and Their Small Perturbations ». Axioms 10, n^o 3 (29 août 2021) : 205. http://dx.doi.org/10.3390/axioms10030205.

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We introduce new necessary conditions for the existence and uniqueness of stationary weak solutions and the existence of the weak solutions for the evolution problem in the system arising from the modeling of the bioconvective flow problem. Our analysis is based on the application of the Galerkin method, and the system considered consists of three equations: the nonlinear Navier–Stokes equation, the incompressibility equation, and a parabolic conservation equation, where the unknowns are the fluid velocity, the hydrostatic pressure, and the concentration of microorganisms. The boundary conditions are homogeneous and of zero-flux-type, for the cases of fluid velocity and microorganism concentration, respectively.

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33

Othman, Saib. « ANALYTIC SOLUTIONS OF PARABOLIC EVOLUTION SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS COUPLED WITH A HYPERBOLIC EQUATION ». Analysis 19, n^o 2 (juin 1999) : 143–64. http://dx.doi.org/10.1524/anly.1999.19.2.143.

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MARKOWICH, P. A., N. MATEVOSYAN, J. F. PIETSCHMANN et M. T. WOLFRAM. « ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION ». Mathematical Models and Methods in Applied Sciences 19, n^o 10 (octobre 2009) : 1929–57. http://dx.doi.org/10.1142/s0218202509003978.

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We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.

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35

Di Marino, Simone, et Alpár Richárd Mészáros. « Uniqueness issues for evolution equations with density constraints ». Mathematical Models and Methods in Applied Sciences 26, n^o 09 (26 juillet 2016) : 1761–83. http://dx.doi.org/10.1142/s0218202516500445.

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In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling crowd motion with hard congestion effects, introduced recently by Maury et al. The monotonicity of the velocity field implies that the [Formula: see text]-Wasserstein distance along two solutions is [Formula: see text]-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an [Formula: see text]-contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only [Formula: see text] as in the standard Fokker–Planck equation.

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36

Papież, Lech S., et George A. Sandison. « A diffusion model with loss of particles ». Advances in Applied Probability 22, n^o 3 (septembre 1990) : 533–47. http://dx.doi.org/10.2307/1427456.

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The dynamical behaviour of particles which undergo diffusion with annihilation is modelled by a parabolic (Fokker–Planck) equation. Fundamental, closed-form solutions of this equation, identified with transition densities of the underlying stochastic process, are calculated by utilizing specific methods of probability measures on functional spaces and evolution semigroups.

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Papież, Lech S., et George A. Sandison. « A diffusion model with loss of particles ». Advances in Applied Probability 22, n^o 03 (septembre 1990) : 533–47. http://dx.doi.org/10.1017/s0001867800019868.

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The dynamical behaviour of particles which undergo diffusion with annihilation is modelled by a parabolic (Fokker–Planck) equation. Fundamental, closed-form solutions of this equation, identified with transition densities of the underlying stochastic process, are calculated by utilizing specific methods of probability measures on functional spaces and evolution semigroups.

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38

Akagi, Goro. « Convergence of functionals and its applications to parabolic equations ». Abstract and Applied Analysis 2004, n^o 11 (2004) : 907–33. http://dx.doi.org/10.1155/s1085337504403030.

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Asymptotic behavior of solutions of some parabolic equation associated with thep-Laplacian asp→+∞is studied for the periodic problem as well as the initial-boundary value problem by pointing out the variational structure of thep-Laplacian, that is,∂φp(u)=−Δpu, whereφp:L2(Ω)→[0,+∞]. To this end, the notion of Mosco convergence is employed and it is proved thatφpconverges to the indicator function over some closed convex set onL2(Ω)in the sense of Mosco asp→+∞; moreover, an abstract theory relative to Mosco convergence and evolution equations governed by time-dependent subdifferentials is developed until the periodic problem falls within its scope. Further application of this approach to the limiting problem of porous-medium-type equations, such asut=Δ|u|m−2uasm→+∞, is also given.

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Giménez, Ángel, Francisco Morillas, José Valero et José María Amigó. « Stability and Numerical Analysis of the Hébraud-Lequeux Model for Suspensions ». Discrete Dynamics in Nature and Society 2011 (2011) : 1–24. http://dx.doi.org/10.1155/2011/415921.

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We study both analytically and numerically the stability of the solutions of the Hébraud-Lequeux equation. This parabolic equation models the evolution for the probability of finding a stressσin a mesoscopic block of a concentrated suspension, a non-Newtonian fluid. We prove a new result concerning the stability of the fixed points of the equation, and pose some conjectures about stability, based on numerical evidence.

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Bachmar, Aziza, et Souraya Boutechebak. « A dynamic problem with wear involving electro-elastic-viscoplastic materials with damage ». Studia Universitatis Babes-Bolyai Matematica 67, n^o 3 (2022) : 653–65. http://dx.doi.org/10.24193/subbmath.2022.3.16.

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"A dynamic contact problem is considered in the paper. The material behavior is described by electro-elastic-viscoplastic law with piezoelectric effects. The body is in contact with damage and an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformations. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a coupled system of an elliptic variational inequality for the displacement, variational equation for the electric potential and a parabolic variational inequality for the damage. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments."

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Aziz, Taha. « On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics : Application of the Deductive Group Theoretical Technique ». Mathematical Problems in Engineering 2021 (14 juin 2021) : 1–10. http://dx.doi.org/10.1155/2021/9974073.

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The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

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Yong, Jiongmin, et Liping Pan. « Quasi-linear parabolic partial differential equations with delays in the highest order spatial derivatives ». Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 54, n^o 2 (avril 1993) : 174–203. http://dx.doi.org/10.1017/s1446788700037101.

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AbstractA class of functional differential equations in some Hilbert space are studied. The results are applicable to many quasi-linear parabolic paratial differential equations with (possibly) countably many discrete delays and finitely many distributed delays in the highest order spatial derivatives. For the linear case, an evolution operator on the underline space H is introduced, via which a variation of constant formula for the solution of the equation in the underline space H is derived. Some spectral properties of the generator of the solution semigroup defined on some appropriate space are discussed as well.

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BARILLON, C., G. M. MAKHVILADZE et V. VOLPERT. « Stability of stationary solutions for a degenerate parabolic system ». European Journal of Applied Mathematics 12, n^o 1 (février 2001) : 57–75. http://dx.doi.org/10.1017/s0956792501004430.

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The paper is devoted to the stability of stationary solutions of an evolution system, describing heat explosion in a two-phase medium, where a parabolic equation is coupled with an ordinary differential equation. Spectral properties of the problem linearized about a stationary solution are analyzed and used to study stability of continuous branches of solutions. For the convex nonlinearity specific to combustion problems it is shown that solutions on the first increasing branch are stable, solutions on all other branches are unstable. These results remain valid for the scalar equation and they generalize the results obtained before for heat explosion in the radially symmetric case [1].

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You, Yuncheng. « Approximate inertial manifolds for nonlinear parabolic equations via steady-state determining mapping ». International Journal of Mathematics and Mathematical Sciences 18, n^o 1 (1995) : 1–24. http://dx.doi.org/10.1155/s0161171295000019.

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For nonlinear parabolic evolution equations, it is proved that, under the assumptions of local Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. It is also shown that the thickness of the exponentially attracting neighborhood of the AIM converges to zero at a fractional power rate as the dimension of the AIM increases. Applications of the obtained results to Burgers' equation, higher dimensional reaction-diffusion equations, 2D Ginzburg-Landau equations, and axially symmetric Kuramoto-Sivashinsky equations in annular domains are included.

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Marinoschi, Gabriela. « Minimal time sliding mode control for evolution equations in Hilbert spaces ». ESAIM : Control, Optimisation and Calculus of Variations 26 (2020) : 46. http://dx.doi.org/10.1051/cocv/2019065.

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This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

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Mesloub, S., A. Bouziani et N. Kechkar. « A Strong Solution of an Evolution Problem with Integral Conditions ». gmj 9, n^o 1 (mars 2002) : 149–59. http://dx.doi.org/10.1515/gmj.2002.149.

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Abstract The paper is devoted to proving the existence and uniqueness of a strong solution of a mixed problem with integral boundary conditions for a certain singular parabolic equation. A functional analysis method is used. The proof is based on an energy inequality and on the density of the range of the operator generated by the studied problem.

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Garcke, Harald. « On Cahn—Hilliard systems with elasticity ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 133, n^o 2 (avril 2003) : 307–31. http://dx.doi.org/10.1017/s0308210500002419.

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Elastic effects can have a pronounced effect on the phase-separation process in solids. The classical Ginzburg—Landau energy can be modified to account for such elastic interactions. The evolution of the system is then governed by diffusion equations for the concentrations of the alloy components and by a quasi-static equilibrium for the mechanical part. The resulting system of equations is elliptic-parabolic and can be understood as a generalization of the Cahn—Hilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it.

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Orlovskii, D. G. « Meaning of evolution of a parameter in an abstract quasilinear parabolic equation ». Mathematical Notes of the Academy of Sciences of the USSR 50, n^o 2 (août 1991) : 847–53. http://dx.doi.org/10.1007/bf01157573.

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Hristov, Jordan. « Heat-balance integral to fractional (half-time) heat diffusion sub-model ». Thermal Science 14, n^o 2 (2010) : 291–316. http://dx.doi.org/10.2298/tsci1002291h.

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The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

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Chen, Ning, et Ji Qian Chen. « Blow-Up of the Solution for some Higher Order Hyperbolic and Neutral Evolution Systems ». Applied Mechanics and Materials 52-54 (mars 2011) : 121–26. http://dx.doi.org/10.4028/www.scientific.net/amm.52-54.121.

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In this paper, we give some results on the blow-up behaviors of the solution to the mixed problem for some higher-order nonlinear hyperbolic and parabolic evolution equation in finite time. By introducing the “ blow-up factor ’’, we get some new conclusions, which generalize some results [4]-[5] , [6] .

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