Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

Ravi P Agarwal,Soha Mohammad Alhumayan,Bessem Samet,Mohamed Jleli

doi:10.3390/math9212765

Ravi P Agarwal, Soha Mohammad Alhumayan + Show 2 more

Open Access

https://doi.org/10.3390/math9212765

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Abstract

In this paper, we study the nonexistence of global weak solutions to higher-order time-fractional evolution inequalities with subcritical degeneracy. Using the test function method and some integral estimates, we establish sufficient conditions depending on the parameters of the problems so that global weak solutions cannot exist globally.

Highlights

We are first concerned with the study of the nonexistence of global weak solutions to time-fractional evolution inequalities of the form:
The study of the nonexistence of global solutions to time-fractional evolution equations and inequalities has been initiated by Kirane and their collaborators
Using the integration by parts rule given by Lemma 1, we define global weak solutions to (1)–(2) as follows

Summary

Introduction

We are first concerned with the study of the nonexistence of global weak solutions to time-fractional evolution inequalities of the form:. In the case where ν = 2, it was shown that under suitable conditions on the initial value u0, if 1 < p ≤ 3, (2)–(4) has no global weak solutions. When ν = 2, it was shown that under suitable conditions for the initial values, if one of the following assumptions is satisfied: (a) N = 2 and 1 < p ≤ k + 1; and (b) N = 2 and p > 1, (2)–(4) has no global weak solutions. The study of the nonexistence of global solutions to time-fractional evolution equations and inequalities has been initiated by Kirane and their collaborators (as can be seen in, e.g., [10,11,12,13,14]).

Preliminaries

Proof of Theorem 1

Proof of Theorem 2

Conclusions