Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems. (English) Zbl 1034.35013

The author considers parabolic equations of the form \[ u_t+Au= F( t,u(\cdot,t))\qquad x\in\Omega,\;t>0 \] where \(\Omega\subset \mathbb{R}^n,\) \(A\) is a second-order self adjoint elliptic operator and \( F(t,\cdot)\) is a superlinear operator. The equation is studied in bounded and unbounded domain. For \(\Omega\not= \mathbb{R}^n\) it is complemented by homogeneous Dirichlet and Neumann boundary conditions. There are obtained a priori estimates for the corresponding solutions in the form \[ \| u(t,u_0)\| \leq C(\| u_0\| ,\delta)\quad \forall\;t<T_{max}(u_0)-\delta \] where \(\delta>0,\) \(u(\cdot, u_0)\) is the solution with initial condition \(u_0\) and \(T_{ max}(u_0)\leq \infty\) is its maximal existence time. Using these estimates the author shows that \(T_{ max}(u_0)\) depends continuously on \(u_0.\) The nonlinearities in the equations are subcritical and they may be nonlocal. The proofs are based on energy, interpolation and maximal regularity estimates. Optimality of the results and some open problems are also discussed.


35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations