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In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sufficient conditions for the globality and for the blow up in finite time of the mild solution. Our approach for the global existence goes back to the Weissler’s technique and for the finite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.

Blow up and globality of solutions for a nonautonomous semilinear heat equation with Dirichlet condition