Fractional-in-Time Semilinear Parabolic Equations and Applications

Abstract

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Local existence and nonexistence for fractional in time weakly coupled reaction-diffusion systems

“…Since we can prove in the same way as in the proof of Theorem 1.3, we leave the proof to readers. Note that we only solve the nonexistence conjecture in [9] when the nonlinear term f is separable with respect to x, t and u.…”

“…Note that if w 2 L r ðXÞ for some 1 r\1, then we can also define in the same way as (1.7) (cf. [9]). Moreover, by definition the operator S a is strongly continuous, i.e., lim…”

“…Fractional in time parabolic equations with nonlinear terms have not been well studied until recently. Gal-Warma [9] has studied the fractional in time scalar problem o a t u ¼ Au þ f ðx; t; uÞ in X Â ð0; TÞ; uðx; 0Þ ¼ u 0 ðxÞ in X;…”

“…Let 1 p\1 and q 1 ; q 2 2 ½1; 1 be given constants. In [9] [20,21], they conjectured the nonexistence of a local in time solution of (1.10) in the super-critical case…”

“…We obtain integrability conditions of ðu 0 ; v 0 Þ which determine the existence/nonexistence of a local in time solution of (1.1). In the proof we combine methods of Gal-Warma [9] and Quittner-Souplet [15,16].…”