Abstract
We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.
Highlights
In this paper we investigate the local and global existence properties of positive solutions of fractional semilinear heat equations of the form ut = αu + f (u), u(0) = φ ∈ L∞(Rn), (1.1)
We present a new dichotomy result for convex nonlinearities f satisfying the ODE
For this class of nonlinearities, we show that all positive solutions of (1.1) blow-up in L∞(Rn) in finite time if and only if f (u) 0+ u2+α/n du = ∞
Summary
In this paper we investigate the local and global existence properties of positive solutions of fractional semilinear heat equations of the form ut = αu + f (u), u(0) = φ ∈ L∞(Rn), where α = − (− )α/2 denotes the fractional Laplacian operator with 0 < α ≤ 2 and f satisfies the monotonicity condition Communicated by Y.Giga.
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