Blow up for \(u_ t- \Delta u=g(u)\) revisited. (English) Zbl 0855.35063

The authors consider the relation between the existence of global classical solutions of the equation \[ u_t- \Delta u= g(u)\text{ in } (0, \infty)\times \Omega,\;u= 0\text{ on } \partial\Omega,\;u(0)= u_0\text{ in } \Omega \tag{1} \] and the existence of weak solutions of the stationary problem (2) \(-\Delta u= g(u)\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\). It is assumed (3) \(g: [0, \infty)\to [0, \infty)\) is a \(C^1\) convex nondecreasing function such that there exists \(x_0\geq 0\) such that \(g(x_0)> 0\) and \(\int_{x_0} ds/g(s)< \infty\). The authors prove the following theorem: If a global classical solution of (1) exists for some \(u_0\in L^\infty(\Omega)\), \(u_0\geq 0\), then there exists a weak solution of (2). As a consequence it follows that if there is no weak solution of (2), then for any initial value \(u_0\in L^\infty(\Omega)\), \(u_0\geq 0\), the solution of (1) blows up in finite time. The authors also prove the following theorem without hypothesis (3): If there exists a weak solution \(w\) of (2), then for any \(u_0\in L^\infty(\Omega)\) with \(0\leq u_0\leq w\), the solution of (1) with \(u(0)= u_0\) is global.


35K57 Reaction-diffusion equations
35J65 Nonlinear boundary value problems for linear elliptic equations