Blow-up phenomena for a class of nonlinear reaction-diffusion equations under nonlinear boundary conditions

ABSTRACT

The paper considers the blow-up solution for the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: $\begin{array}{rl}& \left(h\right(u){)}_t=\mathrm{\nabla }\cdot (\rho \left(|\mathrm{\nabla }u{|}^p\right)|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)+b(x\left)k\right(t\left)f\right(u),(x,t)\in \mathrm{\Omega }\times (0,T),\\ & \rho \left(|\mathrm{\nabla }u{|}^p\right)|\mathrm{\nabla }u{|}^{p-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }n}=g\left(u\right),(x,t)\in \mathrm{\partial \Omega }\times (0,T),\\ & u(x,0)=u_0\left(x\right)\ge 0,x\in \overline{\mathrm{\Omega }},\end{array}$ where p>2 and $\mathrm{\Omega }\subset {\mathbb{R}}^N(N>2)$ is a bounded convex domain with smooth boundary $\mathrm{\partial \Omega }$. By means of constructing auxiliary functions and a first-order differential inequality technique, we obtain the upper and lower bounds on blow-up time under some appropriate assumptions. Moreover, some examples are also presented as application of our main results.

KEYWORDS:

• Reaction-diffusion equations
• blow-up
• lower and upper bounds

AMS CLASSIFICATIONS:

• 35K57
• 35K55
• 35K10

Acknowledgments

The authors are greatly indebted to the reviewer for many helpful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by the opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) (No. KFJJ16-06M). This work is also supported by the key research and development plan of Shanxi Province (No. 201703D221031-1).