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

Solvability for time‐fractional semilinear parabolic equations with singular initial data

“…In Section 5, we establish a nonexistence theorem in $$$ {L}^r\left(\Omega \right),1\le r<\infty $$$. The method used in Section 5 is based on the approach developed in earlier studies 12,17 . Combining existence and nonexistence results, we obtain a necessarily and sufficient condition on the growth of $$$ f $$$ for the existence of a nonnegative solution to () in $$$ {L}^r\left(\Omega \right) $$$.…”

“…the nonexistence of a local-in-time solution to (1.3) was obtained in Suzuki. 12 To the best of our knowledge, Theorem 1.3 is new for general nonlinearities 𝑓 (u).…”

“…For $$$$ f\left(x,t,u\right)=c\left(x,t\right){\left|u\right|}^{p-1}u\kern0.60em \left(x,t,u\right)\in \Omega \times \left[0,\infty \right)\times \mathbb{R}, $$$$ the nonexistence of a local‐in‐time solution to () was obtained in Suzuki 12 . To the best of our knowledge, Theorem 1.3 is new for general nonlinearities $$$ f(u) $$$.…”

“…On the other hand, the nonexistence of a solution to () is not completely understood, apart from the case of pure power nonlinearities $$$ f(u) $$$ (see Suzuki 12 ). The first main result of this work discusses the nonexistence of a nonnegative solution of () which shows the sharpness of the two conditions (i) and (ii) in Proposition 1.2.…”

“…The method used in Section 5 is based on the approach developed in earlier studies. 12,17 Combining existence and nonexistence results, we obtain a necessarily and sufficient condition on the growth of 𝑓 for the existence of a nonnegative solution to (1.1) in L r (Ω). Finally, in Section 6, we prove the uniqueness of a solution of (1.1) in the set K given in Theorem 1.6.…”