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Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions. (English) Zbl 1400.35168

Summary: We study the behaviour of nonnegative global solutions to the quasilinear heat equation with a reaction localized in a ball \[ u_t={\varDelta } u^m+a(x)u^p, \] for \(m>0\), \(0<p\leq \max \{1,m\}\), \(a(x)=1\!\!1 _{B_L}(x)\), \(0<L<\infty \) and \(N\geq 2\). We study when the solutions are bounded or unbounded. In particular we show that the precise value of the length \(L\) plays a crucial role in the critical case \(p=m\) for \(N\geq 3\). We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when \(L=\infty \).

MSC:

35K59 Quasilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
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