Local Existence and Non-Existence for a Fractional Reaction-Diffusion Equation in Lebesgue Spaces

Abstract

We consider the following fractional reaction-diffusion equation u t ( t ) + ∂ t ∫ 0 t g α ( s ) A u ( t − s ) d s = t γ f ( u ) , $$ u_t(t) + \partial_t \int\nolimits_{0}^{t} g_{\alpha}(s) \mathcal{A} u(t-s) ds = t^{\gamma} f(u),$$ where g α (t) = t α−1/Γ(α) (0 < α < 1), f ∈ C([0, ∞)) is a non-decreasing function, γ > −1, and A $\mathcal{A}$ is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in L r (Ω), for non-negative initial data u 0 ∈ L r (Ω), when r > 1 and Ω ⊂ ℝ N is either a smooth bounded domain or the whole space ℝ N . The case r = 1 is also studied.

on a second critical value for the local existence of solutions in lebesgue spaces

“…It is worth mentioning that initial data in the form u 0 = | • | −α χ Ba was used firstly in [10] to show the non-existence of non-negative solution for a system related to (1), and characterizations for parabolic problems related to (1), similar to the results obtained by [6], have been obtained in [1], [5] and for a fractional equation in [4].…”

Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources

“…[32,37]). For results on existence and blow-up in finite time of solutions to (1), one can consult [3,5]. On the other hand, over the last few years the study of existence and stability results in inverse source problems for some kinds of tFrPDEs (including linear and nonlinear types) have drawn the attention of many mathematicians by using various methods, see for instance [14, Section 6.6, p. 247], the introduction of [18] and the references given therein.…”