We consider the Cauchy initial value problem
where is the fractional Laplacian for . We prove that if the initial condition 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
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
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