Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition

Baiping Ouyang,Yan Liu,Zihan Cai,Yiwu Lin

doi:10.1186/s13661-020-01372-x

Abstract

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain (varOmegain mathbb{R}^{n}, ngeq 3) with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.

Highlights

Liu in paper [1] studied the blow-up phenomena for the solution of the following problems: ∂u =um + up uq dx, (x, t) ∈ Ω × 0, t∗, (1.1) ∂tΩ u(x, 0) = f (x) ≥ 0, x ∈ Ω, (1.2)under the Robin boundary condition ∂u + ku = 0,(x, t) ∈ Ω × 0, t∗ . (1.3)
He obtained a lower bound for the blow-up time of the system when the solution blows up
Some authors have started to consider the blow-up of these problems under Robin boundary conditions

Summary

Introduction

1 Introduction Liu in paper [1] studied the blow-up phenomena for the solution of the following problems: He obtained a lower bound for the blow-up time of the system when the solution blows up. In paper [2], the authors studied equations (1.1) and (1.2) subject to either homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. The lower bounds for the blow-up time under the above two boundary conditions were obtained.

Results

Conclusion