Anisotropic Parabolic Equations Research Articles

Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity

We consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.

Removability conditions for anisotropic parabolic equations in a computational validation

The article investigates removability conditions for singularities of anisotropic parabolic equations and in particular for the anisotropic porous medium equation and it aims in the numerical validation of the analytical results. The preconditions on the strength of the anisotropy are analyzed, and the analytical estimates for the growth behavior of the solutions near the singularities are compared with the observed growth in numerical simulations. Despite classical estimates used in the proof, we find that the analytical estimates are surprisingly close to the numerically observed solution behavior.

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Nonnegative weak solutions of anisotropic parabolic equations

Nonnegative weak solutions of anisotropic parabolic equations

Existence of non-negative periodic solutions for a degenerate anisotropic parabolic equation with strongly nonlinear source

Abstract The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.

The fundamental solution and blow-up problem of an anisotropic parabolic equation

This paper is devoted to the study of anisotropic parabolic equation related to the p_{i}-Laplacian with a source term f(u). If f(u)=0, then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy E(0) is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.

On the theory of entropy suband supersolutions of nonlinear degenerate parabolic equations

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem, understood in the sense of Chen-Perthame, is both an entropy sub- and supersolution. It is established that the maximum of entropy subsolutions of the Cauchy problem is also an entropy subsolution of this problem. This result is used to prove the existence of the largest entropy subsolution (and the smallest entropy supersolution). It is also shown that the largest entropy subsolution and the smallest entropy supersolution are also entropy solutions.

Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces

<p style='text-indent:20px;'>We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another conditions we prove unique solvability of the Cauchy problem in the same spaces.</p>

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equationut−∑i=1NDxi(|Dxiu|pi(x,t)−2Dxiu)=f(x,t) in the cylinder Ω×(0,T), where Ω⊂RN, N≥2, is a parallelepiped. The exponents of nonlinearity pi are given Lipschitz-continuous functions. It is shown that if pi(x,t)>2NN+2,μ=supQT⁡maxi⁡pi(x,t)mini⁡pi(x,t)<1+1N,|Dxiu0|max⁡{pi(⋅,0),2}∈L1(Ω),f∈L2(0,T;W01,2(Ω)), then the problem has a unique solution u∈C([0,T];L2(Ω)) with |Dxiu|pi∈L∞(0,T;L1(Ω)), ut∈L2(QT). Moreover,|Dxiu|pi+r∈L1(QT)with some r=r(μ,N)>0,|Dxiu|pi−22Dxiu∈W1,2(QT). The assertions remain true for a smooth domain Ω if pi=2 on the lateral boundary of QT.

Initial-Boundary Value Problems for Anisotropic Parabolic Equations with Variable Exponents of the Nonlinearity in Unbounded Domains with Conditions at Infinity

We deal with the initial-boundary value problems with some restrictions at infinity for linear and nonlinear anisotropic parabolic second-order equations in unbounded domains with respect to the spatial variables. The weak solutions of our problem in Lebesgue and Sobolev spaces with variable exponents is considered. We prove theorems on the existence and uniqueness of the weak solutions using the method based on Saint- Venant principle, and the monotonicity method. Moreover, we obtain estimate of the weak solutions.

On the solvability of the Dirichlet problem for anisotropic parabolic equations in non-convex domains

On the solvability of the Dirichlet problem for anisotropic parabolic equations in non-convex domains

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

INITIAL-BOUNDARY VALUE PROBLEM FOR HIGHER-ORDERS NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENTS OF THE NONLINEARITY IN UNBOUNDED DOMAINS WITHOUT CONDITIONS AT INFINITY

Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|\to +\infty$ (for example, solution's growth restriction as $|x|\to +\infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|\to +\infty$ to solvability of the initial-boundary value problems for parabolic equations considered above. However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity. Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations. In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.

Bounded nonnegative weak solutions to anisotropic parabolic double phase problems with variable growth

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution.

Qualitative analysis for a system of anisotropic parabolic equations with sign‐changing logarithmic nonlinearity

The main purpose of this paper is to discuss the global well‐posedness, asymptotic behavior, and blow‐up of solutions in finite time for a parabolic system of equations driven by the (p(x), q(x))‐Laplacian operator with variable exponents and sign‐changing nonlinearity. More precisely, we consider where is a smooth bounded domain, N ≥ 1, are continuous functions and satisfy some conditions. The main characteristic of this paper is to study the global existence, asymptotic behavior, and blow‐up of solutions to the above system where the reaction terms are sign‐changing and satisfy neither the monotonicity condition nor the Ambrosetti–Rabinowitz condition.

On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation

We give a proof of H ̈older continuity for bounded local weak solutions to the equation ut =\sum_{i=1}^N (|u_{x_i}|^{p_i−2} u_{x_i} )_{x_i} , in Ω × [0, T], with Ω ⊂⊂ R^N under the condition 2 &amp;lt; pi &amp;lt; p(1 + 2/N) for each i = 1, .., N, being p the harmonic mean of the pi's, via recently discovered intrinsic Harnack estimates. Moreover we establish equivalent forms of these Harnack estimates within the proper intrinsic geometry.

Obstacle problem of a nonlinear anisotropic parabolic equation

For sufficiently regular obstacles, we prove existence and uniqueness results for the obstacle problem related to an anisotropic parabolic equation.

Existence and stability of the doubly nonlinear anisotropic parabolic equation

In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated.

On some properties of entropy solutions of degenerate non-linear anisotropic parabolic equations

We prove existence of the largest and the smallest entropy solutions to the Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish the comparison principle in the case when at least one of the initial functions is periodic. In the case when initial function vanishes at infinity (in the sense of strong average) we prove the long time decay of an entropy solution under exact nonlinearity-diffusivity condition.

On the Boundary Value Condition of an Isotropic Parabolic Equation

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, ∏ i = 1 N a i x , t = 0 , x , t ∈ ∂ Ω × 0 , T , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.