Nonlinear Pseudoparabolic Equation Research Articles

Dynamics of a nonlinear pseudo-parabolic equation with fading memory

This paper is concerned with the nonlinear pseudo-parabolic equation with fading memory. First, we prove the existence, uniqueness and continuous dependence of weak solutions when ρ and f have polynomial growth of critical order. Then, we establish the existence and optimal regularity of the global attractor. The result extends and improves some existing results.

Blow-up upper and lower bounds for solutions of a class of higher order nonlinear pseudo-parabolic equations

<abstract><p>In this paper, we study the initial boundary value problem for a class of higher-order nonlinear pseudo-parabolic equations with a memory term. First, the blow-up results of the solution when the initial energy is negative or positive are obtained by using concavity analysis, and an upper bound on the blow-up time $ T^* $ is given. Second, a lower bound on the blow-up time $ T^* $ is obtained by applying differential inequalities when the solutions blow up.</p></abstract>

THE EXISTENCE OF GENERALIZED SOLUTION TO ONEDIMENSIONAL MIXED PROBLEM FOR ONE CLASS OF THIRD ORDER NONILINEAR PSEUDOPARABOLIC EQUATIONS

Abstract. This work is dedicated to the study of existence in small of generalized solution of onedimensional mixed problem for one class of third order nonlinear pseudoparabolic equations. Conception of generalized solution for mixed problem under consideration is introduced. After applying Fourier method, the solution of original problem is reduced to the solution of some countable system of nonlinear inteqro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence theorem in small of generalized solution of the mixed problem is proved by contracted mappings principle.

On one approximate solution of a nonlocal boundary value problem for the Benjamin-Bona-Mahony equation

The paper investigates a non-local boundary value problem for the Benjamin-Bona-Mahony equation. This equation is a nonlinear pseudoparabolic equation of the third order with a mixed derivative. To find a solution to this problem, an algorithm for finding an approximate solution is proposed. Sufficient conditions for the feasibility and convergence of the proposed algorithm are established, as well as the existence of an isolated solution of a non-local boundary value problem for a nonlinear equation. Estimates are obtained between the exact and approximate solution of this problem.

General decay and lifespan of solutions for a system of nonlinear pseudoparabolic equations with viscoelastic term

In this paper, we consider the Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with viscoelastic term. By the Faedo-Galerkin method, we first prove the existence and uniqueness of solution for the problem. Next, we give a sufficient condition to get the global existence and decay of the weak solution. Finally, by the concavity method, we prove the blow-up result of the solution when the initial energy is negative. Furthermore, we establish here the lifespan of the solution by finding the upper bound and the lower bound for the blow-up time.

Reciprocal Bäcklund transformations and travelling wave structures of some nonlinear pseudo-parabolic equations

The equations in which time derivative appears in the highest order derivative term are called pseudo-parabolic equations. In this article, we will propose new singular and dark soliton solutions to a few such equations. We first find the symmetry algebra of such equations, and then by invariant reduction we obtain the exact solutions. Then, we classify the solutions by using their Lie subalgebras. Also, we will discuss the travelling wave solutions by using tanh–coth method. Moreover, we will discuss reciprocal Bäcklund transformations of the conservation laws.

An image encryption algorithm based on heat flow cryptosystems

<abstract><p>Image encryption has been an important research topic in information security. Different from traditional encryption methods, heat flow cryptosystem is a new encryption method. This paper proposes an image encryption algorithm based on heat flow cryptosystem. First, a class of heat flow cryptosystem based on nonlinear pseudo-parabolic equations are given in this paper. Second, a numerical method with high precision namely barycentric Lagrange interpolation collocation method is proposed to solve the nonlinear pseudo-parabolic equation. Third, an image encryption algorithm based on the heat flow cryptosystem is designed, the detailed process of encryption and decryption algorithm is given, the flow diagram of algorithm is showed. Finally, the proposed encryption algorithm is applied to various image with gray and RGB format and compared with the current popular chaotic encryption algorithm. Many indicators such as histograms, information entropy and correlation are used to objectively evaluate the image encryption algorithm. The experimental results show that the proposed image encryption algorithm is better in most indicators and the algorithm is sensitive to the change of key and plaintext.</p></abstract>

General decay for a nonlinear pseudo-parabolic equation with viscoelastic term

Abstract This work is concerned with a multi-dimensional viscoelastic pseudo-parabolic equation with critical Sobolev exponent. First, with some suitable conditions, we prove that the weak solution exists globally. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literature. More precisely, we consider the kernel g : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) g:{[}0,\infty )\hspace{0.33em}\longrightarrow \hspace{0.33em}(0,\infty ) satisfying g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) {g}^{^{\prime} }(t)\leqslant -\xi (t)G(g(t)) , where ξ \xi and G G are functions satisfying some specific properties.

An inverse source problem for a nonlinear pseudoparabolic equation with p-Laplacian diffusion and damping term

In this work, we study the unique solvability of the inverse problem that consists of determining a weak solution u(x, t) to the nonlinear pseudoparabolic equation with p−Laplacian and damping term ut − Δut − div (|∇ u | p −2 ∇u) = γ |u|σ−2 u + f (t)g(x, t) and the coefficient f (t) of right-hand side, which depends on t. Due to the presence of an unknown coefficient in such problems supposes that there is an additional condition along with the initial and boundary conditions. In the present work, an additional condition is represented by the integral overdetermination condition, which represents the average value of a solution tested with some given function over all the domain. The investigating inverse problem considered in two cases: the coefficient of the damping term γ |u|σ−2 u is a positive (nonlinear source term) or negative (an absorption). In both cases, we establish the global and local in time existence and uniqueness of a weak solution to the inverse problem under suitable conditions on the exponents p, σ, the dimension d, and the data of the problem.

General decay and blow-up results for a class of nonlinear pseudo-parabolic equations with viscoelastic term

In this paper, we consider initial boundary value problem of the generalized pseudo-parabolic equation contain viscoelastic term. We establish firstly the local existence of solutions by Banach fixed point theorem. Then we prove blow-up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow-up rate for the weak solution via finding the upper bound and the lower bound for the blow-up times and the upper bound and the lower bound for the blow-up rate. For negative energy, we introduce a new method to prove blow-up results with sharper estimate for upper bound for the blow-up times. Finally, we prove the global existence of the solution and a general decay estimate under some suitable assumptions.

Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified groups

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in Ruzhansky and Suragan (J Differ Eq 262:1799–1821, 2017) for stratified groups.

Solvability of an initial-boundary value problem for a nonlinear pseudoparabolic equation with degeneration

This article is devoted to the solvability of degenerate nonlinear equations of pseudoparabolic type. Such problems appear naturally in physical and biological models. The article aims to study the solvability in the classes of regular solutions of (all derivatives generalized in the sense of S.L. Sobolev included in the equation) initial-boundary value problems for differential equations. For the problems under consideration, we have found conditions on parameters ensuring the existence of solutions and we have proved existence and uniqueness theorems. The main method for proving the solvability of boundary value problems is the regularization method.

On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions

<p style='text-indent:20px;'>In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.</p>

Spectral discretizations analysis with time strong stability preserving properties for pseudo-parabolic models

In this work, we study the numerical approximation of the initial-boundary-value problem of nonlinear pseudo-parabolic equations with Dirichlet boundary conditions. We propose a discretization in space with spectral schemes based on Jacobi polynomials and in time with robust schemes attending to qualitative features such as stiffness and preservation of strong stability for a more correct simulation of non-regular data. Error estimates for the corresponding semidiscrete Galerkin and collocation schemes are derived. The performance of the fully discrete methods is analyzed in a computational study.

Cauchy problem for inhomogeneous fractional nonclassical diffusion equation on the sphere

Pseudo-parabolic equation on spheres have many important applications in physical phenomena, oceanography and meteorology, geophysics. The main purpose of this paper is to prove the existence and unique solution of the nonlinear pseudo-parabolic equation on the sphere. To do this, we used some analysis of Fourier series associated with several evaluations of the spherical harmonics function. Some of the upper and lower bounds of the Mittag-Lefler functions are also used. This result is one of the first studies of fractional nonclassical diffusion equation on the sphere.

SOLVABILITY OF A NONLINEAR INVERSE PROBLEM FOR A PSEUDOPARABOLIC EQUATION WITH P-LAPLACIAN

Inverse problems of determining the right-hand side of a differential equation arise in the mathematical modeling of many physical phenomena, when an external source or some of its parameters acting to the motion of the process are unknown or unacceptable for measurement, for example, the source is in a high-temperature environment or underground, etc. This paper deals to study the solvability of an inverse problem for a nonlinear pseudo-parabolic equation (sometimes they called Sobolev-type equations) with p-Laplacian and damping term with variable exponent. The inverse problem consists of determining a coefficient of the right-hand side depending only on time. Additional information for this investigated inverse problem is given as an integral overdetermination condition. Under the suitable conditions on the exponents and on the data the global and local in time existence of a weak solutions to the inverse problem are established. The existence of a weak solution proved by the Faedo-Galerkin method. The global and local in time a priori estimates for the Galerkin approximations are obtained. On the basis of a priori estimates and by using compactness theorems and the monotonicity method, the convergence of the Galerkin approximations to the solution of the initial inverse problem is proved. Key words: inverse problem, pseudoparabolic equations, existence, weak solution.

Mixed Problem for a Higher-Order Nonlinear Pseudoparabolic Equation

We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters.

A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results

In this paper, by applying the Faedo-Galerkin approximation method and using basic concepts of nonlinear analysis, we study the initial-boundary value problem for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions. It consists of two main parts. Part 1 is devoted to proof of the unique existence of a weak solution by establishing an approximate sequence u m based on a N -order iterative scheme in case of f ∈ C N 0,1 × 0 , T ∗ × ℝ N ≥ 2 , or a single-iterative scheme in case of f ∈ C 1 Ω ¯ × 0 , T ∗ × ℝ . In Part 2, we begin with the construction of a difference scheme to approximate u m of the N -order iterative scheme, with N = 2 . Next, we present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

General decay and blow‐up results for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions

This paper is devoted to the study of a nonlinear pseudoparabolic equation in an annular with Robin–Dirichlet conditions. At first, by applying the standard Faedo–Galerkin method, we prove existence and uniqueness results. Next, using concavity method, we prove blow‐up results for solutions when the initial energy is nonnegative or negative, then we also establish the lifespan for the equation via finding the upper bound and the lower bound for the blow‐up times. Finally, we give a sufficient condition for the global existence and decay of weak solutions.

LOCAL EXISTENCE AND EXPONENTIAL DECAY OF SOLUTIONS FOR A NONLINEAR PSEUDOPARABOLIC EQUATION WITH VISCOELASTIC TERM

In this paper, we investigate an initial boundary value problem for a nonlinear pseudoparabolic equation. At first, by applying the Faedo-Galerkin, we prove local existence and uniqueness results. Next, by constructing Lyapunov functional, we establish a sufficient condition to obtain the global existence and exponential decay of weak solutions.