Problem For Nonlinear Parabolic Equation Research Articles

Stochastic Model of the Cauchy–Neumann Problem for Nonlinear Parabolic Equations

Stochastic Model of the Cauchy–Neumann Problem for Nonlinear Parabolic Equations

BLOW-UP PROBLEMS FOR FUJITA-TYPE PARABOLIC SYSTEM INVOLVING TIME-DEPENDENT COEFFICIENTS ON GRAPHS

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

Differentiation Functors and Category Interpretation of Optimality Conditions

Operator derivatives are determined as functors. Necessary optimality conditions with category interpretation are proved for abstract optimization control problems. Finite dimensional extremum problems and an optimization control problem for nonlinear parabolic equation with state constraints are considered as examples.

Terminal value problem for nonlinear parabolic equation with Gaussian white noise

<abstract><p>In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>.</p></abstract>

On Weak Solution of Cauchy Problem for Nonlinear Parabolic Equations

In this paper, we investigate weak solutions of Cauchy problem for nonlinear parabolic equations. By using Galerkin approximation, the existence and uniqueness of weak solutions are proved.

Second-Order Regularity for Parabolic p-Laplace Problems

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

Blow-up problems for nonlinear parabolic equations on locally finite graphs

Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut=Δu + f(u) on G. The blow-up phenomenons for ut=Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

Monte Carlo method for solution of initial–boundary value problem for nonlinear parabolic equations

In this paper we will consider the initial–boundary value problem for a parabolic equation with a polynomial non-linearity relative to the unknown function. First we will derive a probabilistic representation of our problem. The representation of the solution of this problem is given in the form of a mathematical expectation, which is determined based on trajectories of branching processes. Under the assumption of the existence of the solution, an unbiased estimator is built using trajectories of a branching process. We will use a mean value theorem to write out a special integral equation, that equates the value of the unknown function u(x,t) with its integral over a spheroid and balloid with center at the point (x,t). A probabilistic representation of the solution to the problem in the form of mathematical expectation of some random variables is obtained. This probabilistic representation uses a branching process whose trajectories are used in the contraction of an unbiased estimator for the solution. The derived unbiased estimator has a finite variance, and is built up from trajectories of branching processes with a finite average number of branches. Finally, the results of numerical experiments and application to the practical problems are discussed.

Identification of the initial condition in backward problem with nonlinear diffusion and reaction

In this paper, we study the backward in time problem for nonlinear parabolic equations associated with nonlinear diffusion coefficient. The problem is ill-posed in the sense of Hadamard. Under an a priori assumption on the exact solution in Gevrey space, we propose a new regularization method for stabilising the ill-posed problem. Our results extend some earlier works on backward problems for nonlinear parabolic equations.

Boundary-Value Problems for Nonlinear Parabolic Equations with Delay and Degeneration at the Initial Time

We study boundary-value problems with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. The a priori estimates of this solution are obtained.

Optimal control problem for systems governed by nonlinear parabolic equations without initial conditions

An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control in the case of final observation is proved.

Extinction of solutions of higher order parabolic equations with double nonlinearity and degenerate absorption potential

We study the extinction property of solutions to the Cauchy–Dirichlet problem for nonlinear parabolic equations of the order 2m with absorption potential in a semibounded cylinder (0,+∞) × Ω, where Ω is a bounded domain in ℝN, N ≥ 1. The sufficient conditions ensuring the extinction of a solution in a finite time, which depend on N,m, and q (where q is a parameter of the homogeneous nonlinearity in the main part of the equation), are obtained.

Cauchy problem for nonlinear parabolic equations with a gradient term

We consider the Cauchy problem of the nonlinear diffusive Hamilton–Jacobi equationut−Δum=|Duq|σ, where m>0. The existence and nonexistence of local and global solutions are studied by a priori estimates and compactness methods. We prove that the Cauchy problem of this equation has no nontrivial nonnegative global solutions if 0<σ<σ⁎ (here σ⁎=Nm+2qN+1); and there exist nontrivial global solutions for small initial values if σ>σ⁎. Moreover, we obtain the global existence of solutions in the limiting case and the exact L∞-estimates for local and global solutions.

Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations

We consider the Cauchy problem for nonlinear parabolic equations with functional dependence. We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equatio

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \partial_t u=\Delta u+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad in \quad{\bf R}^N, $$ and assume that the solution $u$ behaves like the Gauss kernel as $t\to\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $\varphi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to the large class of nonlinear parabolic equations.

Approximation methods in optimal control problems for nonlinear infinite-dimensional systems

Some notions related to approximate solutions and to the approximation of extremum problems for nonlinear infinite-dimensional systems are proposed. Optimization problems for nonlinear parabolic equations with a fixed terminal state and on an infinite time interval, as well as for singular stationary systems with phase constraints, are illustrated by several examples.

On a backward nonlinear parabolic equation with time and space dependent thermal conductivity: Regularization and error estimates

Abstract. We study the inverse problem for nonlinear parabolic equation with variable thermal conductivity a ( x , t ) ${a(x,t)}$ which is a ill-posed problem. We regularize this problem by using the truncated integral method and then obtain the error estimate between the exact solution and the regularized solution. A numerical experiment is given for illustrating our method.

A Regularization of the Backward Problem for Nonlinear Parabolic Equation with Time-Dependent Coefficient

We study the backward problem with time-dependent coefficient which is a severely ill-posed problem. We regularize this problem by combining quasi-boundary value method and quasi-reversibility method and then obtain sharp error estimate between the exact solution and the regularized solution. A numerical experiment is given in order to illustrate our results.

Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space

We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space. We state sufficient conditions for uniqueness or nonuniqueness of bounded solutions, depending on the behavior of the density at infinity. Nonuniqueness relies on the prescription at infinity of suitable conditions of Dirichlet type, and possibly inhomogeneous.