Abstract
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared in the pointwise sense to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
Highlights
Introduction and main resultsNonlinear evolution problems involving fractional Laplacian describing the anomalous diffusion have been extensively studied in the mathematical and physical literature, see [43] for the Cauchy problem (1.1)–(1.2), and [9,10,11,12] for other examples of problems and for extensive list of references
This paper is a straightforward generalization of extensively studied classical nonlinear heat equation, see [40], to the case of nonlocal but linear diffusion operators defined by fractional powers of Laplacian
Existence of global-in-time solutions for initial data in a suitable critical Morrey space is shown in Proposition 2.3
Summary
Nonlinear evolution problems involving fractional Laplacian describing the anomalous diffusion (or the α-stable Levy diffusion) have been extensively studied in the mathematical and physical literature, see [43] for the Cauchy problem (1.1)–(1.2), and [9,10,11,12] for other examples of problems and for extensive list of references. One of these models is the following initial value problem for the reaction-diffusion equation with the anomalous diffusion ut = −(−Δ)α/2u + |u|p−1u, Rd × (0, ∞),.
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