Blowing-up solutions for time-fractional equations on a bounded domain

This paper proposes initial-boundary value problems for time-fractional analogs of Kuramoto-Sivashinsky, Korpusov-Pletner-Sveshnikov, Cahn-Allen, and Hoff equations due to a bounded domain. Adequate conditions for the blowing-up of solutions in limited time of previously mentioned conditions are displayed. The Pohozhaev nonlinear capacity strategy is considered. Illustrative examples are given for each of the investigated equations. (2010) Mathematics Subject Classifications: 34A08; 34B25; 34B15.


Introduction
Fractional calculus works on the powers of the differential equations that are not numbers such that most wonders in science and designing are communicated by partial differential equations. Fractional calculus was initiated as a pure mathematical aspect in the middle of the 19th century. 1,2 The concept of fractional or noninteger order derivation and integration can be followed back to the beginning of numbers order calculus itself. 3 For the most part, physical marvel might depend on its current state and on its chronicled states, which can be displayed effectively by applying the hypothesis of derivatives and integrals of fractional order. 4 Due to this, several analytical techniques are used to derive exact, explicit, and numerical solutions of nonlinear fractional partial differential equations, where the modeling of physical phenomena is very interest to many scientists and researchers up to now. 5 With these achievements, we study time-fractional equations for funding blowing-up solutions by using the Pohozhaev nonlinear capacity method [6][7][8][9][10][11] ; more absolutely, on the choice of test functions according agreeing to initial and boundary conditions beneath thought for the time-fractional equations. In 2021, Alsaedi et al. gave a simple case of the analysis of a rough blow-up, that is, the case where the solution tends to infinity as s ! t 8 on ½0, ,, more exactly, when tends to infinity as s ! t 8 for the given function F. 6 The aim of this paper is to study the blowing-up solutions of the following the time-fractional equations: TF2. Time-fractional Korpusov-Pletner-Sveshnikov equation TF3. Time-fractional Cahn-Allen equation TF4. Time-fractional Hoff equation for v 2 j = : (0, ,), s.0, here j i (i = 1, 2, 3), non-zero constants with initial conditions where z 0 is a given function. This work is devoted to blowing-up solutions of time-fractional analogues of the above equations. The approach to the problem is based on the Pohozhaev nonlinear capacity method; more precisely, on the choice of test functions according to initial and boundary conditions under consideration. The solutions for nonlinear partial differential equations has attracted a large number of researchers, as many papers have emerged around this study (see  ). We donate a straightforward case of the investigation of a harsh blow-up, that is, the case where the solution tends to infinity as s ! t 8 on j more precisely, for the given function F, the integral tends to infinity as s ! t 8 .
The rest of the paper can be outlined concisely as follows: ''Preliminaries'' section contains some definitions and properties of fractional order integral and differential operators that will be used later. In the ''Blowingup solutions of the time-fractional equations'' section the Pohozhaev nonlinear capacity method has been applied to above equation and illustrates the obtained results by some examples at the end of each section.

Preliminaries
For real-valued function v 2 L 1 (J), the fractional integral of Riemann-Liouville is defined by G is the Euler gamma function 16 and ''Ã'' is the convolution operation. The Sobolev space is defined for the function v by (Kilbas et al. 16 ) : Let v Ã K mÀz (s) 2 W m, 1 (J) and m = ½z + 1, z.0. The Riemann-Liouville fractional derivative D z + t 1 of order z.0 (m À 1\z\m, m 2 N) is defined as The Caputo fractional derivative ∂ z + t 1 of order If v 2 C m (J), then the Caputo fractional derivative ∂ z + t 1 of order z 2 R is defined as Let a given function z be monotone. We consider the FDE The Theorem 2.2 assure the blow-up of solutions to (9).
The solution of problem (9) blows-up in a finite time 18 is lim s! z(s) = +', whenever z 0 .0.

Blowing-up solutions of the time-fractional equations
The time-fractional generalized Kuramoto-Sivashinsky equation In this section we consider the time-fractional generalized Kuramoto-Sivashinsky equation (1) where j 1 , j 2 , j 3 are the parameters and 0\z ł 1 is the fractional order. The problem (1) is also called KdV-Burgers-Kuramoto equation with time-fractional derivative as it may be a generalization of the taking after well-known equations. The equation (1) coincides with The classical fractional Burgers equation take the form whenever j 1 .0 and j 2 = j 3 = 0 in (1); The classical fractional KdV equation take the form whenever j 1 = j 3 = 0 and j 2 = 1 in (1); The classical fractional Kuramoto-Sivashinsky equation take the form (1), and z = 1. We investigate the question of the blow-up of a classical solution of problem (1). Assume that 2 C 4 ( j). Using integrating by parts from multiplication the time-fractional generalized Kuramoto-Sivashinsky equation (1) and j, we get where and 8v 2 j, s 2 ½0, t, Consider monotonically nondecreasing function j(v), and satisfy the following properties Then we have The Ho¨lder inequality implies that Then, expression (14) takes the form H(z(,, s), j(,)) À H(z(0, s), j(0)): , the function c satisfy conditions (16) and the solution z of the equation (1) belongs to C 1, 4 v, s ( j 3 (0, t)). If F(s)À 1 ø 0, for all s.0, and W(0).0, then W(s) ! + ', 8s ! t 8 , where t 8 satisfies estimate (10).

Proof.
Obviously whereW(s) = q 2 W(s). Since the functionW(s) is an upper solution of equation (9), soW(s) ! + ' for where inequality (10) hols for t 8 . Whereupon  The time-fractional Korpusov-Pletner-Sveshnikov equation In this section we consider the time-fractional Korpusov-Pletner-Sveshnikov equation (2), here, j 1 , j 2 are the parameters and z is the fractional order with 0\z ł 1. The equation (2) is called the Korpusov-Pletner-Sveshnikov equation with time-fractional derivative as it is a generalization of the following wellknown equations. We study the question of the blowup of a classical solution z 2 C 1, 2 v, s ( j 3 ½0, t) of the problem (2). Assume that j 2 C 2 ( j) and the solution z 2 C 1, 2 v, t, ( j 3 ½0, t) of the problem (2) exists. Multiplying the time-fractional Korpusov-Pletner-Sveshnikov equation (2) by j and integrating by parts, we obtain where F(s) = H(z(,, s), j(,)) À H(z(0, s), j(0)), Consider monotonically nondecreasing function j(v) with and following properties Then we have where z(v, s) = z(v, s) + j 1 j 2 : Form the Ho¨lder inequality, we get Then, expression (17) takes the form where Proof. Obviously ∂ z + 0, vW (s) øW 2 (s), whereW(s) = q 2 W(s). Since the functionW(t) is an upper solution of equation (9), thereforeW(s) ! + ' for s ! t 8 , where estimate (10) holds for t 8 . Whereupon W(s) ! + ' for The time-fractional Cahn-Allen equation In this section we consider the time-fractional Cahn-Allen equation (3), where z is the fractional order with 0\z ł 1. The equation (3) is called the Cahn-Allen equation with time-fractional derivative as it is a generalization of the following well-known equations. We investigate the question of the blow-up of a classical solution z 2 C 1, 2 v, s ( j 3 ½0, t) of problem (3). Assume that j 2 C 2 ( j) and the solution z 2 C 1, 2 v, s ( j 3 ½0, t) of problem (3) exists. Multiplying the time-fractional Cahn-Allen equation (3) by z v j, we have where H(z(v,s),j(v))= À 0:5∂ a +0,s z 2 (v,s)j(v)+0:5z 2 v (v,s)j(v) À 0:25U 4 (v,s)j(v)+0:5z 2 (v,s)j(v): Consider the monotonically nondecreasing function (v), such that j 0 (v) ø 0, 8v 2 j, and satisfy the following properties Then we have where z(v, s) = z(v, s) À 1. By employing the Ho¨lder inequality, we have we also get À 0:5 Then, expression (20) takes the form where The time-fractional Hoff equation In this section we consider the time-fractional Hoff equation where z is the fractional order with 0\z ł 1.
We investigate the question of the blow-up of a classical solution z 2 C 1, 2 v, s ( j 3 ½0, t) of the problem (4). Assume that j 2 C 2 ( j) and the solution z 2 C 1, 2 v, s ( j 3 ½0, t) of the problem (4) exists. Multiplying the time-fractional Hoff equation (4) by j and integrating by parts, we have and let the function j satisfy conditions (26). If F(s) Àq 1 ø 0, (8s.0), and W(0).0, then W(s) ! + ' for each s ! t 8 , where t 8 satisfies estimate (10).

Conclusion
We donate a straightforward case of the investigation of a harsh blow-up, that is, the case where the solution tends to infinity as s ! t 8 on j more precisely, when for the given function F, the integral (6) tends to infinity as s ! t 8 .

Author contributions
AB: Actualization, formal analysis, methodology, initial draft, validation, and investigation. MKAK: Methodology, actualization, validation, investigation, formal analysis, and initial draft. MB: Methodology, actualization, validation, investigation, formal analysis, and initial draft. MES: Validation, actualization, formal analysis, methodology, investigation, simulation, initial draft, software, and was a major contributor in writing the manuscript. XGY: Methodology, actualization, validation, investigation, formal analysis, and initial draft. All authors read and approved the final manuscript.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD
Mohammed KA Kaabar https://orcid.org/0000-0003-2260-0341 Availability of data and materials Data sharing not applicable to this article as no datasets we're generated or analyzed during the current study.