Abstract
In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sucient conditions for the globality and for the blow up infinite time of the mild solution. Our approach for the global existence goes back to the Weissler's technique and for the nite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.
Highlights
Consider a semilinear heat equations of the typeAfter de pioneering works of Kaplan [17] and Fujita [11, 12], many authors have studied global existence and blow up in finite time of positive solutions for semilinear heat equations (1) (with and without Dirichlet conditions) when k ≡ 1 and A = ∆, the Laplacian operator, for several types of nonlinearities η
Consider a semilinear heat equations of the type ∂u(t, x)= k(t)Au(t, x) + h(t)η(u), (t, x) ∈ (0, T ] × D, (1)∂t u(0, x) = f (x), x ∈ D, u(t, x) = 0, (t, x) ∈ (0, T ] × Dc, where D ⊂ Rd is an open set, k, h : [0, ∞) → [0, ∞) are continuous and not identically zero functions, f is a continuous function on D, A is the infinitesimal generator of a symmetric Levy process and the nonlinearity η(u) is assumed to satisfy η(0) = 0 and η(u) > 0 for u > 0
It is well known that solutions of many differential equations of the type (1) with or without Dirichlet conditions, can become unbounded in finite time
Summary
After de pioneering works of Kaplan [17] and Fujita [11, 12], many authors have studied global existence and blow up in finite time of positive solutions for semilinear heat equations (1) (with and without Dirichlet conditions) when k ≡ 1 and A = ∆, the Laplacian operator, for several types of nonlinearities η. D ⊂ Rd, d ≥ 1, is a bounded nonempty open set, A is the infinitesimal generator of a symmetric Levy process {Zt}t≥0, β > 1 is a constant, the initial value f is a nonnegative function in the space C0(D) of continuous functions on D vanishing on Dc and the time dependent coefficients k, h : [0, ∞) → [0, ∞) are continuous and not identically zero. |||Ψu − Ψv||| ≤ βRβ−1 h(s) u(s, ·) − v(s, ·) ∞ds
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