Instantaneous blow-up of semilinear non-autonomous equations with fractional diffusion

Date

2017-05-02

Authors

Villa-Morales, Jose

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Publisher

Texas State University, Department of Mathematics

Abstract

We consider the Cauchy initial value problem ∂/∂t u (t, x) = k(t)∆α u (t, x) + h(t)ƒ(u(t, x)), u(0, x) = u0(x), where ∆α is the fractional Laplacian for 0 < α ≤ 2. We prove that if the initial condition u0 is non-negative, bounded and measurable then the problem has a global integral solution when the source term ƒ is non-negative, locally Lipschitz and satisfies the generalized Osgood's condition ∫∞∥u0∥∞ ds/ƒ(s) ≥ ∫∞0 h(s)ds. Also, we prove that if the initial data is unbounded then the generalized Osgood's condition does not guarantee the existence of a global solution. It is important to point out that the proof of the existence hinges on the role of the function h. Analogously, the function k plays a central role in the proof of the instantaneous blow-up.

Description

Keywords

Generalized Osgood's condition, Semilinear equations, Fractional diffusion, Instantaneous blow-up

Citation

Villa-Morales, J. (2017). Instantaneous blow-up of semilinear non-autonomous equations with fractional diffusion. <i>Electronic Journal of Differential Equations, 2017</i>(116), pp. 1-10.

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Attribution 4.0 International

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