Qualitative properties for solutions of semilinear heat equations with strong absorption (Q1399326)

This paper considers the following nonlinear heat equation \[ u_t-\Delta u +b(x,t)|u|^{p-1}u=0 \quad \text{ in } \mathbb R^N\times(0,\infty), \] where \(0<p\leq 1\), \(b\geq 0\), and the solution \(u\) possesses the initial value \(u_0\) satisfying \(0<u_0(x)\leq f_0\exp(A|x|^{\alpha})\) with \(f_0>0\), \(A>0\) and \(\alpha\geq 2\). The main results include: (i) If \(b\) satisfies certain non-degenerate condition, then the solutions imposed exponential growth restriction obey the comparison principle; (ii) the solutions exist globally; and (iii) if in addition \(0<p<1\), then such solutions will become compactly supported for \(t>0\), and vanish in finite time. The necessity of the condition on \(b\) is also exhibited.