Quasilinear Pseudoparabolic Equations Research Articles

Inverse coefficient problem for quasilinear pseudo-parabolic equation by fourier method

In this research, we consider a coefficient problem of an inverse problem of a quasilinear pseudo-parabolic equation with nonlocal boundary condition. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also we consider numerical solution for this inverse problem by using linearization and finite difference method.

Weak solutions of impulsive pseudoparabolic equations with an infinitesimal transition layer

We study the multi-dimensional initial–boundary value problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term, which models a non-instantaneous impulsive impact. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instantaneous impulsive impact. We prove that the infinitesimal transition layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of weak solutions of the original problem converges to the weak solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter equation inherits the full information about the profile of the original non-instantaneous impulsive impact.

Strong solutions of impulsive pseudoparabolic equations

We study the two-dimensional Cauchy problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term endowed with periodic initial data and periodicity conditions. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which models an instantaneous impulsive impact. We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of strong solutions of the original problem converges to the strong solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter is derived based on the microstructure of the transition layer profile.

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.

IDENTIFICATION OF THE RIGHT HAND SIDE OF A QUASILINEAR PSEUDOPARABOLIC EQUATION WITH MEMORY TERM

The study of equations of mathematical physics, including inverse problems, is relevant today. This work is devoted to the fundamental problem of studying the solvability and qualitative properties of the solution of the inverse  problem  for  a  quasilinear  pseudoparabolic  equation (also called Sobolev-type equations) with memory term. To date, studies of direct and inverse problems for a pseudoparabolic equations are rapidly developing in connection with the needs of modeling and control of processes in thermal physics, hydrodynamics, and mechanics of a continuous medium. The pseudoparabolic equations similar to those considered in this work arise in the description of heat and mass transfer processes, processes of non-Newtonian fluids motion, wave processes, and in many other areas. The main types of the inverse problems are: boundary, retrospective, coefficient and geometric. The boundary and retrospective inverse problems lead to the study of linear problems. In turn, the statements related to the study of coefficient and geometric types bring to the nonlinear problems. Coefficient inverse problems are divided into two main groups: coefficient inverse problems, where the unknown is a function of one or several variables, and finite-dimensional coefficient inverse problems. In this article the existence and uniqueness of a weak and strong solution of the inverse problem in a bounded domain are proved by the Galerkin method. Also we used Sobolev’s embedding theorems, and obtained a priori estimates for the solution. Moreover, we get local and global theorems on the existence of the solution. Key words: Pseudoparabolic equation, inverse problem, existence, uniqueness, local solvability, global solvability, non-local condition.

Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation \t\t\tut−△ut−△u−div(|∇u|2q∇u)=up\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{t}-\\triangle u_{t}-\\triangle u-\\operatorname{div}\\bigl(| \\nabla u|^{2q}\\nabla u\\bigr)=u^{p} $$\\end{document} which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016), where the blow-up phenomena and the lifespan for the initial energy J(u_{0})<0 were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J(u_{0})<0.

Blow-up phenomena for a nonlinear pseudo-parabolic equation with nonlocal source

We discuss a class of quasi-linear pseudo-parabolic equation with nonlocal source u t − Δ u t − ∇ ⋅ | ∇ u | 2 q ∇ u = u p ( x , t ) ∫ Ω k ( x , y ) u p + 1 ( y , t ) d y x , y ∈ Ω , t ∈ ( 0 , T 0 ] , where q ≤ p and 0 < q < n − 2 2 . By establishing the criterions for blow-up, we determine the upper bounds for blow-up time under not only q < p and non-positive initial energy but also q = p and negative initial energy. The results show that the upper bound for blow-up time under q < p is different from it under q = p . Moreover, we also determine the lower bound for blow-up time.

Weak generalized and numerical solution for a quasilinear pseudo-parabolic equation with nonlocal boundary condition

This paper investigates the one dimensional mixed problem with nonlocal boundary conditions, for the quasilinear parabolic equation. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, convergence of the weak generalized solution, and also continuous dependence upon the data of the solution are shown by using the generalized Fourier method. We construct an iteration algorithm for the numerical solution of this problem.

On solvability of the first initial-boundary value problem for quasilinear pseudoparabolic equations

In the paper, the solvability of the first initial-boundary value problem for a quasilinear pseudoparabolic equationut-Δut-∂∂xibijx,t,uuxj+cix,t,uuxi+ϑx,t,u=fx,twith increasing with respect to ζ functions bijx,t,ζ,cix,t,ζ and ϑx,t,ζ was investigated. The existence and uniqueness of regular solutions are proven.

A PRIORI L2ERROR ANALYSIS FOR AN EXPANDED MIXED FINITE ELEMENT METHOD FOR QUASILINEAR PSEUDO-PARABOLIC EQUATIONS

Based on an expanded mixed finite element method, we consider the semidiscrete approximations of the solution u of the quasilinear pseudo-parabolic equation defined on <TEX>${\Omega}{\subset}R^d$</TEX>, <TEX>$1{\leq}d{\leq}3$</TEX>. We construct the semidiscrete approximations of <TEX>${\nabla}u$</TEX> and <TEX>$a(u){\nabla}u+b(u){\nabla}u_t$</TEX> as well as u and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of <TEX>${\nabla}u$</TEX> and <TEX>$a(u){\nabla}u+b(u){\nabla}u_t$</TEX> as well as u in <TEX>$L^2$</TEX> normed space.

Some problems for nonlinear pseudoparabolic equations in nontube domains

We prove existence and uniqueness of a weak solution to the first initial-boundary value problem for some class of quasilinear pseudoparabolic equations in nontube domains. Also, we study unique solvability in these domains for the variational inequality connected with the above class of equations.

Solution to a transmission problem for quasilinear pseudoparabolic equations by the Rothe method

In this paper, we deal with a transmission problem for a class of quasilinear pseudoparabolic equations. Existence, uniqueness and continuous dependence of the solution upon the data are obtained via the Rothe method. Moreover, the conver- gence of the method and an error estimate of the approximations are established.

Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities

The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂ t u is replace by ∂ t b ( u ) , with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitz-continuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be Lipschitz continuous.

On the mixed problem for quasilinear pseudoparabolic equation

In this paper the multidimensional mixed problem for the quasilinear pseudoparabolic equation ut-Lxu-εLxut=f(t,x,u) is considered. Lx is a differential operator, which composes (with boundary operator) a self adjoint operator. An existence, uniqueness and also continuous dependense on the small parameter ε>0 of generalized solution is proved. The estimation of the difference of exact and approximate solutions is obtained

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer ΠT=ℝn×[0,T] for the equationut=cGΔut+Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC1(ℝ+) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈Cx,t2,1(ΠT) with the properties:\(\varphi ^1 (u(x,t)) \leqslant M_1 (1 + |x|^2 )\),\(|u_t (x,t)| \leqslant M_2 (1 + |x|^2 )^\beta (\beta > 0)\).

On a mixed problem for systems of definite quasilinear partial differential equations with deviating argument

For a mixed problem for a system of definite quasilinear pseudoparabolic equations with deviating argument, we prove a theorem on differential inequalities and existence of a unique regular solution and a comparison theorem and give sufficient conditions of existence of solutions with constant sign.

Fourier Spectral Methods for Pseudoparabolic Equations

Fourier spectral methods of Galerkin and collocation type for quasilinear pseudo-parabolic equations are presented. Convergence with spectral accuracy is proved for both Galerkin and collocation approximations. Unconditionally stable discretization in time by explicit one-step methods is analyzed. Fast transform methods to compute the nonlinear terms are proposed.