Blow-up at space infinity for a quasilinear parabolic equation with space-dependent reaction

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We consider a nonnegative solution $u$ of the Cauchy problem for a quasilinear parabolic equation $u_t=\Delta u^m+\mu(x)u^p$ with the initial data $u_0(x)\,(\not\equiv 0)$ satisfying $\| \tilde\mu u_0\|_{L^{\infty}(\mathbf R^N)}<\infty$, where nonnegative function $\mu(x)$ satisfies some condition and $\tilde\mu=\mu^{1/(p-1)}$. We give sufficient conditions on $u_0$ for a weighted solution $\tilde\mu u$ to blow up at space infinity and for a direction $\psi \in \mathbf S^{N-1}$ to be a blow-up direction of $\tilde\mu u$. We also show that such a weighted solution $\tilde\mu u$ blows up completely at the blow-up time of $\tilde\mu u$.