On critical exponents for semilinear heat equations with nonlinear boundary conditions

Abstract

This paper deals with the blow-up properties of solutions to semilinear heat equation \(u_t - \Delta u = u^p {\text{ in }}R_ + ^N {\text{ }}x (0,T)\) with the nonlinear boundary condition \( - \frac{{\partial u}}{{\partial x}} = u^q for{\text{ }}x1 = t \in (T)\). It has been proved that if max(p,q)≤1,every nonnegative solution is global. When min(p, q)>1 by letting α=1/p−1 and β=1/2(q−1) it follows that if max (α,β)≤N/2,all nontrivial non-negative solutions are nonglobal, whereas if max(α,β)< N/2,there exist both global and non-global solutions. Moreover, the exact blow-up rates are established.