Well-posedness results for a class of semilinear time-fractional diffusion equations

Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients

“…Firsts, we consider the following definition. Definition 4.9 (Continuation, see [8,37]). Given a mild solution u ∈ X αs ((0, T ]; L 2 (Ω)) of (P), we say that u is a continuation of u in (0, T ] for T > T if it is satisfying u ∈ X αs ((0, T ]; L 2 (Ω)) is a mild solution of (P) for all t ∈ (0, T ], u (x, t) = u(x, t) whenever t ∈ [0, T ], x ∈ Ω.…”

“…The next results are on global existence or non-continuation by a blow-up and depend continuously on the initial data. Definition 4.11 (Maximal existence time, see [8,37]). Let u(x, t) be a weak solution of (P).…”

“…(ii) If there exists T ∈ (0, ∞) such that u(x, t) exists for 0 ≤ t < T , but does not exist at t = T , then T max = T . Definition 4.12 (Finite time blow-up, see [8,37]). Let u(x, t) be a weak solution of (P).…”

Analysis of large time asymptotics of the fourth‐order parabolic system involving variable coefficients and mixed nonlinearities

“…Here, $$$ {2}_{\chi}&#x0005E;{\ast}:&#x0003D; \frac{2N}{N-2\chi } $$$, $$$ \chi \in \left(-\frac{N}{2},\frac{N}{2}\right) $$$, and the positive parameters $$$ {\left\{{c}_{ij},{d}_{ij}\right\}}_{i,j&#x0003D;1}&#x0005E;M $$$ serve as scaling factors. These nonlinear sources appear in some physical phenomena (see, e.g., previous studies [1–12]).…”

Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces

“…Further, the papers [7], [4], [5], [23] are devoted to studying of abstract parabolic equation with a fractional time derivative in Banach spaces.…”