Nonlinear fractional differential equations of Sobolev type

Abstract

Communicated by S. G. GeorgievSobolev type nonlinear equations with time fractional derivatives are considered. Using the test function method, limiting exponents for nonexistence of solutions are found.

On the absence of global weak solutions for some differential inequalities of Sobolev type in an exterior domain

“…Further, let Ω be the exterior domain of R N , N ≥ 3, given by (1). Let us denote by F the solution to the exterior problem −ΔF = 0, x ∈ Ω,…”

“…For other related works, see, for example, reference herein. 1,11,13,14 The study of nonexistence criteria in an exterior domain has been considered by many mathematicians. Bidaut-Véron 3 studied the problem…”

Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative

“…However, most of the results for fractional differential or integral equations mentioned above are concerned with the Riemann-Liouville fractional derivative or the Caputo fractional derivative (see for example [15][16][17][18][19][20]). There are few works on fractional differential equations involving the Hadamard fractional derivative, which is presented as a quite different kind of weakly singular kernel.…”

Instantaneous blow‐up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field

“…Later, the effect of instantaneous blow‐up for linear and nonlinear Sobolev‐type equation has not been considered since researchers were interested in sufficient conditions of the existence of solutions. In Alsaedi et al, the global insolvability of Sobolev‐type equations with fractional time derivatives was examined. A new result obtained in Korpusov states that the equation which does not contain singular coefficients of the form | x | − α or t − β , whose initial functions belong to the class , does not have local‐in‐time solutions under the condition …”

“…Later, the effect of instantaneous blow-up for linear and nonlinear Sobolev-type equation has not been considered since researchers were interested in sufficient conditions of the existence of solutions. In Alsaedi et al, 12 the global insolvability of Sobolev-type equations with fractional time derivatives was examined. A new result obtained in Korpusov 13 states that the equation…”