Semilinear Pseudo-parabolic Equation Research Articles

Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity

In this paper, a semilinear pseudo-parabolic equation with a general nonlinearity is considered. A new finite time blow-up criterion for this problem is established. Moreover, the upper bound for the blow-up time is also derived. The results partially generalize some recent ones obtained in [Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Applied Mathematics Letters, 83(2018), 176-181] by Runzhang Xu et al.

A note on the life span of semilinear pseudo-parabolic equation

We consider the Cauchy problem for the semilinear pseudo-parabolic equation with initial data with compact support. We find the life span of the solution, when the power exponent p is smaller than the Fujita exponent.

A semilinear pseudo-parabolic equation with initial data non-rarefied at ∞

In this paper we study the Cauchy problem for a semilinear pseudo-parabolic equation with initial data non-rarefied at ∞. Our interest lies in the discussion of the effect of the non-rarefied factors on the bow-up phenomenon and the life span of the solution.

Global Existence and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Pseudo-parabolic Equations

In this paper, we study the initial-boundary value problem for the semilinear pseudo-parabolic equations ut − ΔXut − ΔXu = |u|p−1u, where X = (X1,X2,…,Xm) is a system of real smooth vector fields which satisfy the Hormander’s condition, and ΔX = $${{\rm{\Delta}}_X} = \sum\nolimits_{j = 1}^m {X_j^2} $$ is a finitely degenerate elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.

The Well-Posedness of Solution to Semilinear Pseudo-parabolic Equation

In this paper, we use the Green’s function method to get the pointwise convergence rate of the semilinear pseudo-parabolic equations. By using this precise pointwise structure and introducing negative index Sobolev space condition on the initial data, we release the critical index of the nonlinearity for blowing up. Our result shows that the global existence does not only depend on the nonlinearity but also the initial condition.

Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity \begin{document}$u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$\end{document} , where \begin{document}$X=(X_1, X_2, ··· , X_m)$\end{document} is an infinitely degenerate system of vector fields, and \begin{document}$\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$\end{document} is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincare inequality, we obtain the global existence and blow-up at \begin{document}$+∞$\end{document} of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy

We establish a new finite time blowup theorem for the solution to the initial boundary value problem of a class of semilinear pseudo-parabolic equations at high initial energy level. And we also estimate the upper bound of the blowup time.

A note on blow-up of solution for a class of semilinear pseudo-parabolic equations

In this short note, we revisit the blow-up of solution for the initial boundary value problem of semilinear pseudo-parabolic equations with low/critical initial energy stated in Xu and Su (2013) [4], and amend the proofs of the original paper.

Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term

ABSTRACTIn this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term where is a bounded domain in () with a Dirichlet boundary condition. Under suitable assumptions on the initial data and the relaxation function g, we obtain the global existence and finite time blow-up of solutions with initial data at low energy level (i.e. ), by using the Galerkin method, the concavity method and an improved potential well method involving time t. We also derive the upper bounds for the blow-up time. Finally, we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.

Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration

In this paper, we study the semilinear pseudo-parabolic equations $$\displaystyle u_{t} - \triangle _{{\mathbb {B}}}u - \triangle _{{\mathbb {B}}}u_{t} = \left| u\right| ^{p-1}u$$ on a manifold with conical singularity, where $$\triangle _{{\mathbb {B}}}$$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $$x_{1} = 0$$ . Firstly, we discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy $$J(u_{0})<d$$ , the solution is global in time with $$I(u_{0}) >0$$ or $$\displaystyle \Vert \nabla _{{\mathbb {B}}}u_{0}\Vert _{L_{2}^{\frac{n}{2}}({\mathbb {B}})} = 0$$ and blows up in finite time with $$I(u_{0}) < 0$$ ; for the critical initial energy $$J(u_{0}) = d$$ , the solution is global in time with $$I(u_{0}) \ge 0$$ and blows up in finite time with $$I(u_{0}) < 0$$ . The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.

Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source

This paper considers the following semilinear pseudo-parabolic equation with a nonlocal source: $$ u_{t}-\triangle u_{t}-\triangle u=u^{p}(x,t) \int_{\Omega}k(x,y)u^{p+1}(y,t)\,dy, $$ and it explores the characters of blow-up time for solutions, obtaining a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, we investigate a nonblow-up criterion and compute an exact exponential decay.

Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we consider the periodic problem for semilinear heat equation and pseudo-parabolic equation with logarithmic source. After establishing the existence of positive periodic solutions, we discuss the instability of such solutions.

Addendum to “Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations” [J. Func. Anal. 264 (12) (2013) 2732–2763

In this remark, we correct the proof of the asymptotic behavior of solution for the initial boundary value problem of semilinear pseudo-parabolic equations with critical initial energy obtained in Xu and Su (2013) [1].

Small perturbation of a semilinear pseudo-parabolic equation

This paper is concerned with large time behavior of solutions for the Cauchy problem of a semilinear pseudo-parabolic equation with small perturbation. It is revealed that small perturbation may develop large variation of solutions with the evolution of time, which is similar to that for the standard heat equation with nonlinear sources.

Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we study the initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, we obtain the existence of global solution, blow-up at +∞ and behavior of vacuum isolation of solutions. On the other hand, the asymptotic behavior of solutions is also discussed. Our result implies that the polynomial nonlinearity is important for the solutions of such kinds of semilinear pseudo-parabolic equations to be blow-up in finite time.

Blow-up in non-autonomous semilinear pseudoparabolic equations

We study the blow-up property for weak solutions to the Cauchy problem of non-autonomous semilinear pseudoparabolic equations. Given the growth bound of the non-autonomous coefficient, the Fujita-type critical exponent is obtained.

Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations

In this paper, we study a class of semilinear pseudo-parabolic equations. By introducing a family of potential wells, we prove the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with initial energy J(u0)⩽d. Moreover, we obtain finite time blow-up with high initial energy J(u0)>d by comparison principle.

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.

Cauchy problems of semilinear pseudo-parabolic equations

This paper concerns with the Cauchy problems of semilinear pseudo-parabolic equations. After establishing the necessary existence, uniqueness and comparison principle for mild solutions, which are also classical ones provided that the initial data are appropriately smooth, we investigate large time behavior of solutions. It is shown that there still exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations.