On the critical set for discrete Laplacian parabolic equations with polynomial-type reactions

Abstract

In this paper, we study long-time behaviours of solutions to the discrete reaction-diffusion equations $u_t={\mathrm{\Delta }}_{\omega }u+\psi (t)f(u)$ with nontrivial and nonnegative initial data, under the mixed boundary conditions. In particular, the function f satisfies $\alpha (u^q+u^p)\le f(u)\le \beta (u^q+u^p)$ for some constants $0<\alpha \le \beta$ and positive real numbers p and q. The purpose of this paper is to introduce a critical set ${\mathrm{\Lambda }}_{\psi }$ depending on the nonnegative continuous function ψ. Also, we discuss about Fujita-type blow-up, which depends on the critical set ${\mathrm{\Lambda }}_{\psi }$.

Keywords:

• Parabolic equation
• discrete Laplacian
• polynomial
• Fujita-type blow-up
• critical set

AMS CLASSIFICATIONS:

• 39A12
• 35F31
• 35K91
• 35K57

Acknowledgements

The second author was supported by BK21 PLUS SNU Mathematical Sciences Division. The authors would like to express thanks to anonymous reviewers for their excellent suggestions

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Min-Jun Choi was supported by BK21 PLUS SNU Mathematical Sciences Division.