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

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.

Keywords

• Generalized Osgood's condition
• semilinear equations
• fractional diffusion
• instantaneous blow-up