A necessary and sufficient condition for the existence of global solutions to reaction-diffusion equations on bounded domains

Abstract

Abstract The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations u t = Δ u + ψ ( t ) f ( u ) , in Ω × ( 0 , ∞ ) , $$ u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), $$ under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case f ( u ) = u p $f(u)=u^{p}$ . As a matter of fact, we prove: there is no global solution for any initial data if and only if ∫ 0 ∞ ψ ( t ) f ( ∥ S ( t ) u 0 ∥ ∞ ) ∥ S ( t ) u 0 ∥ ∞ d t = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 ( Ω ) . $$ \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} $$ Here, ( S ( t ) ) t ≥ 0 $(S(t))_{t\geq 0}$ is the heat semigroup with the mixed boundary condition.

Keywords

• Blow-up
• Global existence
• Mixed boundary
• Nonlinear parabolic equation