Solutions Of Parabolic Equations Research Articles

A complete characterization of Fujita's blow-up solutions for discrete p-Laplacian parabolic equations under the mixed boundary conditions on networks

In this paper, we study long-time behaviors of solutions to the discrete p-Laplacian parabolic equations with time-dependent reactionsut=Δp,ωu+ψ(t)|u|q−1u with nontrivial and nonnegative initial data, under the mixed boundary conditions, where p≥2, q>0, and ψ is a positive continuous function. The goal of this paper is to characterize completely the parameters p, q, and function ψ to see when the solutions are Fujita's blow-up solutions, general blow-up solutions, or global solutions.

On Large-Time Behavior of Solutions of Parabolic Equations

We study the stabilization of solutions of the Cauchy problem for second-order parabolic equations depending on the behavior of the lower-order coefficients of equations at the infinity and on the growth rate of initial functions. We also consider the stabilization of solution of the first boundary-value problem for a parabolic equation without lower-order coefficients depending on the domain Q where the initial function is defined for t =0.
 In the first chapter, we study sufficient conditions for uniform in x on a compact K RN stabilization to zero of the solution of the Cauchy problem with divergent elliptic operator and coefficients independent of t and depending only on x. We consider classes of initial functions:
 
 bounded in RN,
 with power growth rate at the infinity in RN,
 with exponential order at the infinity.
 
 Using examples, we show that sufficient conditions are sharp and, moreover, do not allow the uniform in RN stabilization to zero of the solution of the Cauchy problem.
 In the second chapter, we study the Cauchy problem with elliptic nondivergent operator and coefficients depending on x and t. In different classes of growing initial functions we obtain exact sufficient conditions for stabilization of solutions of the corresponding Cauchy problem uniformly in x on any compact K in RN. We consider examples proving the sharpness of these conditions.
 In the third chapter, for the solution of the first boundary-value problem without lower-order terms, we obtain necessary and sufficient conditions of uniform in x on any compact in Q stabilization to zero in terms of the domain RN \ Q where Q is the definitional domain of the initial function for t =0. We establish the power estimate for the rate of stabilization of the solution of the boundary-value problem with bounded initial function in the case where RN \ Q is a cone for t =0.

Numerical solutions of fractional parabolic equations with generalized Mittag–Leffler kernels

AbstractIn this article, we investigate the generalized fractional operator Caputo type (ABC) with kernels of Mittag–Lefller in three parameters and its fractional integrals with arbitrary order for solving the time fractional parabolic nonlinear equation. The generalized definition generates infinitely many problems for a fixed fractional derivative α. We utilize this operator with homotopy analysis method for constructing the new scheme for generating successive approximations. This procedure is used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the ℏ‐curves as well as by calculating the residual error and the results were accurate. Based on the experiment, we verify the existence of the solution for the new parameters. Depending on these results, this treatment can be used to find approximate solutions to many fractional differential equations.

The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation

We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii)~the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.

Hölder continuity of singular parabolic equations with variable nonlinearity

Abstract In this paper we obtain the local Hölder regularity of the weak solutions for singular parabolic equations with variable exponents. The proof is based on DiBenedetto’s technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result.

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.

Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in

We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when i.e., they behave like where is the maximal ergodic constant and is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem.

A third-order accurate in time method for boundary layer flow problems

The boundary layer flow problem arises in numerous industrial applications. As a result, it has received considerable attention over the last five decades which has involved developing numerical procedures to approximate the solution of time-dependent parabolic and first-order hyperbolic partial differential equations (PDEs). In this paper, we develop a method that guarantees third-order temporal accuracy. Stability conditions are derived using Von Neumann stability analysis that guarantee convergence of the proposed algorithm. In addition, we present a consistency analysis. The performance of the proposed algorithm is demonstrated for linear and nonlinear parabolic PDEs that extend the models for the Stokes first and second problems by incorporating the effect of heat transfer using viscous dissipation and thermal radiation. A comparison of performance in terms of convergence rate and estimation accuracy is shown.

Asymptotic method of moving planes for fractional parabolic equations

In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions.We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal parabolic problems with more general operators and more general nonlinearities.

The applications of Sobolev inequalities in proving the existence of solution of the quasilinear parabolic equation

The aim of this paper is to show some applications of Sobolev inequalities in partial differential equations. With the aid of some well-known inequalities, we derive the existence of global solution for the quasilinear parabolic equations. When the blow-up occurs, we derive the lower bound of the blow-up solution.

Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks

We study the long-time behaviour of phenotype-structured models describing the evolutionary dynamics of asexual populations whose phenotypic fitness landscape is characterised by multiple peaks. First we consider the case where phenotypic changes do not occur, and then we include the effect of heritable phenotypic changes. In the former case the model is formulated as an integrodifferential equation for the phenotype distribution of the individuals in the population, whereas in the latter case the evolution of the phenotype distribution is governed by a non-local parabolic equation whereby a linear diffusion operator captures the presence of phenotypic changes. We prove that the long-time limit of the solution to the integrodifferential equation is unique and given by a measure consisting of a weighted sum of Dirac masses centred at the peaks of the phenotypic fitness landscape. We also derive an explicit formula to compute the weights in front of the Dirac masses. Moreover, we demonstrate that the long-time solution of the non-local parabolic equation exhibits a qualitatively similar behaviour in the asymptotic regime where the diffusion coefficient modelling the rate of phenotypic change tends to zero. However, we show that the limit measure of the non-local parabolic equation may consist of less Dirac masses, and we provide a sufficient criterion to identify the positions of their centres. Finally, we carry out a detailed characterisation of the speed of convergence of the integral of the solution (i.e. the population size) to its long-time limit for both models. Taken together, our results support a more in-depth theoretical understanding of the conditions leading to the emergence of stable phenotypic polymorphism in asexual populations.

Entropy solutions for nonlinear parabolic equations in Musielak Orlicz spaces without Delta_2-conditions

In this paper, we are interested in results concerning entropy solutions for nonlinear parabolic equations in Musielak Orlicz spaces without Δ2-condition.

Time Analyticity for Inhomogeneous Parabolic Equations and the Navier–Stokes Equations in the Half Space

We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (2020)] and Dong-Zhang [J. Funct. Anal. (2020)], where the authors proved the time analyticity of solutions to the homogeneous heat equation and the Navier-Stokes equations in the whole space.

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems

In this paper, the existence and uniqueness of the weak solution for a linear parabolic equation with conformable derivative are proved, the existence of weak periodic solutions for conformable fractional parabolic nonlinear differential equation is proved by using a more generalized monotone iterative method combined with the method of upper and lower solutions. We prove the monotone sequence converge to weak periodic minimal and maximal solutions. Moreover, the conformable version of the Lions-Magness and Aubin–Lions lemmas are also proved.

Correction to: Traveling wave solutions for degenerate nonlinear parabolic equations

Correction to: Traveling wave solutions for degenerate nonlinear parabolic equations

Compact almost automorphic solutions for semilinear parabolic evolution equations

ABSTRACT In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the existence of at least one bounded solution on the right half line. More specifically, we improve the assumptions in [Cieutat P, Ezzinbi K. Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities. J Funct Anal. 2011;260(9):2598–2634.], we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order ) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). For that purpose we distinguish two cases, and . The main difficulty in this work, is to prove the existence of at least one solution with relatively compact range while the forcing term is not necessarily bounded. Moreover, we propose to study a large class of reaction-diffusion problems with unbounded forcing terms.

Behavior of blow-up solutions for quasilinear parabolic equations

We study the quasilinear parabolic equation (|u|q − 1u)t − Δpu = 0 in a multidimensional domain (0; T) × Ω under the condition u(t; x) = f(t; x) on (0; T) × 𝜕Ω, where the boundary function f blows-up at a finite time T, i.e., f(t; x) → ∞ as t → T. For p ⩾ q > 0 and the boundary function f with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where p > q to p = q is investigated. A general approach within the method of energy estimates to such problems is described.

Approximation of backward stochastic partial differential equations by a splitting-up method

This paper develops a splitting-up method to solution of backward stochastic parabolic partial differential equations. Both splitting equations are simpler for the numerical computations. Two different approximation schemes are given, and their convergence results are proved. The first-order convergence rate is proved in the linear case.

Traveling wave solutions for degenerate nonlinear parabolic equations

We consider the traveling wave solutions of the degenerate nonlinear parabolic equation $$u_{t}=u^{p}(u_{xx}+u)$$ which arises in the model of heat combustion, solar flares in astrophysics, plane curve evolution problems and the resistive diffusion of a force-free magnetic field in a plasma confined between two walls. We also deal with the equation $$v_{\tau }=v^{p}(v_{xx}+v-v^{-p+1})$$ related with it. We first give a result on the whole dynamics on the phase space $${\mathbb {R}}^{2}$$ with including infinity about two-dimensional ordinary differential equation that introduced the traveling wave coordinates: $$\xi = x-ct$$ by applying the Poincare compactification and dynamical system approach. Second, we focus on the connecting orbits on it and give a result on the existence of the weak traveling wave solutions with quenching for $$c>0$$ and $$p\in 2{\mathbb {N}}$$ . Moreover, we give the detailed information about the asymptotic behavior of $$u(\xi )$$ , $$u'(\xi )$$ , $$v(\xi )$$ and $$v'(\xi )$$ for $$p\in 2{\mathbb {N}}$$ . In the case that $$p \in 2 {\mathbb {N}}+1$$ , it is too complicated to determine the dynamics near the singularities on the Poincare disk, however, we classify the connecting orbits and corresponding traveling wave solutions and obtain their asymptotic behavior.

Decay of Nonnegative Solutions of Singular Parabolic Equations with KPZ-Nonlinearities

The Cauchy problem for quasilinear parabolic equations with KPZ-nonlinearities is considered. It is proved that the behavior of the solution as $$t \to \infty $$ can change substantially as compared with the homogeneous case if the equation involves zero-order terms. More specifically, the solution decays at infinity irrespective of the behavior of the initial function and the rate and character of this decay depend on the conditions imposed on the lower order coefficients of the equation.