Instantaneous blow-up of semilinear non-autonomous equations with fractional diffusion
Date
2017-05-02
Authors
Villa-Morales, Jose
Journal Title
Journal ISSN
Volume Title
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.
Rights
Attribution 4.0 International