Semilinear Pseudo-parabolic Equation Research Articles

Determination of a time dependent source in semilinear pseudoparabolic equations with Caputo fractional derivative

This paper devoted to study unique solvability of inverse source problem for a semilinear time fractional pseudoparabolic equation perturbed by a damping term. Inverse problem consists of recovering a solely time dependent coefficient of right-hand side under a measurement in an integral form. The damped term acts in the equation as a nonlinear source or as an absorption, depending on its sign of coefficient, where it is positive or negative, respectively. In these both cases, we have established sufficient conditions on data, where the inverse problem has a global or local in time unique weak and strong solution.

The Asymptotic Behavior and Blow-Up Rate of a Solution with a Lower Bound on the Highest Existence Duration for Semi-Linear Pseudo-Parabolic Equations

This note addresses the initial-boundary value problem for a class of semi-linear pseudo-parabolic equations defined on a smooth bounded domain, with an emphasis on determining the asymptotic behavior and blow-up rate of the solution. Our analysis considers both low-initial energy and critical-initial energy cases, with a specific focus on establishing a lower bound on the maximal existence time of the solutions to this problem.

Uniform Stabilization and Asymptotic Behavior with a Lower Bound of the Maximal Existence Time of a Coupled System’s Semi-Linear Pseudo-Parabolic Equations

Uniform Stabilization and Asymptotic Behavior with a Lower Bound of the Maximal Existence Time of a Coupled System’s Semi-Linear Pseudo-Parabolic Equations

Existence and blow-up of solutions for a class of semilinear pseudo-parabolic equations with cone degenerate viscoelastic term

Existence and blow-up of solutions for a class of semilinear pseudo-parabolic equations with cone degenerate viscoelastic term

Blow-up and Bounds of Solutions for a Class of Semi-Linear PseudoParabolic Equations with p(. )-Laplacian Viscoelastic Term

Blow-up and Bounds of Solutions for a Class of Semi-Linear PseudoParabolic Equations with p(. )-Laplacian Viscoelastic Term

Strong solutions of a semilinear impulsive pseudoparabolic equation with an infinitesimal initial layer

We study the multi-dimensional initial-boundary value problem for the semilinear pseudoparabolic equation with a regular nonlinear minor term, which, in general, may be superlinear. This term models a non-instantaneous but a very rapid absorption with q(x)-growth. The minor term depends on a positive integer parameter n and, as n→+∞, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as n→+∞, and that the family of regular weak solutions to 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, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the initial layer is governed by the linear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the initial 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 absorption. In general, the research is devoted to pseudoparabolic equations with measure data depending on an unknown solution.

Blow-up time of solutions to a class of pseudo-parabolic equations

In this paper, we study the Dirichlet problem for a semilinear pseudo-parabolic equation. By using the energy estimates and ordinary differential inequalities, we studied the upper and lower bounds of blow-up time of the solutions. The results of this paper extend and complete the results on this model.

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.

Bounds for blow-up solutions of a semilinear pseudo-parabolic equation with a memory term and logarithmic nonlinearity in variable space

In this article, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a logarithmic nonlinear source term \[ u_{t}-\Delta u_{t}+\int _{0}^{t}g( t-s) \Delta u( x,s) \mathrm {d}s-\Delta u\]\[=|u|^{p(\cdot ) -2}u\ln (|u|), \]with a Dirichlet boundary condition. Under appropriate assumptions about the relaxation function $g$, the initial data $u_{0}$ and the function exponent $p$, we not only set the lower bounds for the blow-up time of the solution when blow-up occurs, but also by assuming that the initial energy is negative, we give a new blow-up criterion and an upper bound for the blow-up time of the solution.

Exponential growth of solution for a couple of semi-linear pseudo-parabolic equations with memory and source terms

This work is concerned with coupled semi-linear pseudo-parabolic equations with memory terms in both equations, associated with the homogeneous Dirichlet boundary condition. We show that the solution grows exponentially under specific conditions regarding the relaxation functions and initial energy. In order to prove the result, we use the energy method based on the construction of a suitable Lyapunov function.The most important behavior of the evolution system is the exponential growth phenomena because of its wide range of applications in modern science, such as chemistry, biology, ecology, and other areas of engineering and physical sciences.

Global solutions of a fractional semilinear pseudo-parabolic equation with nonlocal source

In this paper, the initial boundary value problem for a fractional nonlocal semilinear pseudo-parabolic equation is established. Firstly, we get the local solution by the standard Galerkin method and the priori estimates. Next, by applying potential well argument, the existence and uniqueness of the global solution are proved for initial energy .

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.

Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

Global well-posedness for fractional Sobolev-Galpern type equations

<p style='text-indent:20px;'>This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.</p>

Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source

We consider a semilinear pseudo-parabolic equation with nonlinearities of variable exponent subject to Dirichlet boundary conditions We prove that the nonnegative classical solutions blow-up in finite time with arbitrary positive initial energy and suitable large initial values. Also, employing a differential inequality technique, we obtain an upper bound for blow-up time if and the initial value satisfies some conditions.

Asymptotic analysis for a pseudo-parabolic equation with nonstandard growth conditions

The main goal of this work is to study a semilinear pseudo-parabolic equation with nonstandard growth conditions. For initial energy , by using the generalized potential well method, we establish a sharp threshold result on global existence or blow-up weak solution. On the other hand, we also show the asymptotic behavior of solutions and the blow-up time estimates. The optimal estimations: the asymptotic behavior are obtained when the blow-up occurs. Moreover, for high initial energy , we illustrate the conditions on global existence or blow-up.

Semilinear pseudo-parabolic equations on manifolds with conical singularities

<p style='text-indent:20px;'>This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.</p>

Well-posedness of the solution of the fractional semilinear pseudo-parabolic equation

This article concerns the Cauchy problem for the fractional semilinear pseudo-parabolic equation. Through the Green’s function method, we prove the pointwise convergence rate of the solution. Furthermore, using this precise pointwise structure, we introduce a Sobolev space condition with negative index on the initial data and give the nonlinear critical index for blowing up.

Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation

Abstract In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.

Global well-posedness for a nonlocal semilinear pseudo-parabolic equation with conical degeneration

This paper deals with a class of nonlocal semilinear pseudo-parabolic equation with conical degenerationut−△But−△Bu=|u|p−1u−1|B|∫B|u|p−1udx1x1dx′, on a manifold with conical singularity, where △B is Fuchsian type Laplace operator with totally characteristic degeneracy on the boundary x1=0. By using the modified method of potential well with Galerkin approximation and concavity, the global existence, uniqueness, finite time blow up and asymptotic behavior of the solutions will be discussed at the low initial energy J(u0)<d and critical initial energy J(u0)=d, respectively. Furthermore, we investigate the global existence and finite time blow up of the solutions with the high initial energy J(u0)>d by the variational method. Especially, we also derive the threshold results of global existence and nonexistence for the solutions at two different initial energy levels, i.e. low initial level and critical initial level.