Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions ☆

Abstract This paper is mainly concerned with the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary-value problem of the fast diffusive polytropic filtration equation with reaction sources.

Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources

The authors of this paper study some singular
phenomena (blowing-up or vanishing in finite time) of solutions to
the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear
diffusion equation involving the $p(x,t)$-Laplace operator
and a nonlinear source. The variable exponent $p(x,t)$ leads to the
failure of some techniques, such as upper-lower solutions technique
and the scaling method etc., in studying the problem; it also leads
to the lack of some valuable properties such as the monotonicity of
the energy integral etc. The authors construct a suitable control
functional, improve the regularity of the approximate solutions and
obtain a new energy inequality to prove that the solution of the
problem with a positive initial energy blows up in finite time.
Furthermore, under some appropriate conditions, the authors study
the vanishing property and the extinction rate estimate of the
solutions to the problem by establishing some inequalities the
solutions satisfy. It is worth pointing out that the results are
obtained with the assumption that $p_{t}(x,t)$ is only negative
and integrable which is weaker than those the most of the other
papers required.

Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)$ -Laplace operator and a non-local term.

The aim of this paper was to study vanishing and blowing-up properties of the solutions to a homogeneous initial Dirichlet problem of a nonlinear diffusion equation involving the p(x)-Laplace operator and a nonlinear source. The authors point out that the results obtained are not trivial generalizations of similar problems in the case of constant exponent because the variable exponent p(x) brings some essential difficulties such as the failure of upper and lower solution method and scaling technique, the existence of a gap between the modular and the norm. To overcome these difficulties, the authors have to improve the regularity of solutions, to construct a new control functional and apply suitable embedding theorems to prove the blowing-up property of the solutions. In addition, the authors utilize an energy estimate method and a comparison principle for ODE to prove that the solution vanishes in finite time. At the same time, the critical extinction exponents and an extinction rate estimate to the solutions are also obtained.

Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplace operator and nonlinear sources

In this note we illuminate that the small condition on initial data u0 in Theorem 4.1 of Yin and Jin (Math. Meth. Appl. Sci. 2007; 30(10):1147–1167) can be removed for the case p−1<q<1. Precise decay estimates of solution are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.

A note on extinction for fast diffusive p-Laplacian with sources

We discuss and determine the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary value problem of the fast diffusive p-Laplacian with sources. Copyright © 2007 John Wiley & Sons, Ltd.

Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources

This paper is concerned with the p-Laplacian equation with singular sources which is allowed to change sign in a ball. The singularity may occur at the zero points of solutions and on the boundary. Using the upper and lower solutions method, we establish the existence of positive radial solutions for the problem considered. Copyright © 2006 John Wiley & Sons, Ltd.

Positive radial solutions of p-Laplacian equation with sign changing nonlinear sources

This paper is concerned with the existence of periodic solutions of the Nicholson's blowflies model with Newtonian diffusion. By constructing some suitable Lyapunov functionals and combining with Leray–Schauder fixed point theorem, we establish the existence of nonnegative time periodic solutions. Copyright © 2009 John Wiley & Sons, Ltd.

Existence of time periodic solutions for the Nicholson's blowflies model with Newtonian diffusion

This paper is concerned with the existence, uniqueness and attractability of time periodic solutions of a Cahn–Hilliard type equation with periodic gradient-dependent potentials and sources. Copyright © 2009 John Wiley & Sons, Ltd.

Chunhua Jin

Papers

Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources

Positive radial solutions of p-Laplacian equation with sign changing nonlinear sources

Existence of time periodic solutions for the Nicholson's blowflies model with Newtonian diffusion

A Cahn–Hilliard type equation with periodic gradient‐dependent potentials and sources

Critical exponents of evolutionary p-laplacian with interior and boundary sources