Electron. J. Differential Equations, Vol. 2017 (2017), No. 116, pp. 1-10.

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

Jose Villa-Morales

Abstract:
We consider the Cauchy initial value problem
$$\displaylines{ \frac{\partial }{\partial t}u(t,x) =k(t)\Delta _{\alpha}u(t,x)+h(t)f(u(t,x)), \cr u(0,x) = u_0(x), }$$
where $\Delta _{\alpha }$ is the fractional Laplacian for $0<\alpha \leq 2$. We prove that if the initial condition $u_0$ is non-negative, bounded and measurable then the problem has a global integral solution when the source term f is non-negative, locally Lipschitz and satisfies the generalized Osgood's condition
$$ \int_{\|u_0\|_{\infty }}^{\infty }\frac{ds}{f(s)}\geq \int_0^{\infty}h(s)ds. $$
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.

Submitted September 5, 2016. Published May 2, 2017.
Math Subject Classifications: 35K05, 45K05, 60G52, 34G20.
Key Words: Generalized Osgood's condition; semilinear equations; fractional diffusion; instantaneous blow-up.

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José Villa-Morales
Departamento de Matemáticas y Física
Universidad Autónoma de Aguascalientes
Av. Universidad No. 940, Cd. Universitaria
Aguascalientes, Ags., C.P. 20131, Mexico
email: jvilla@correo.uaa.mx

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