Blowing-up solutions of the time-fractional dispersive equations

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow. The Korteweg-de Vries equation [KV95] is well known in different fields of science and technology, it reads u t + uu x + u xxx = 0.
(1.2) In [BBM72], Benjamin, Bona and Mahony proposed the following equation to describe long waves on the water surface u t − u txx + uu x = 0. (1.3) In [Ros86] Rosenau suggested the following equation to describe waves on "shallow" water: u t + u txxxx + u x + uu x = 0.
(1.4) In [Ost78] Ostrovsky derived an equation for weakly nonlinear surface and internal waves in a rotating ocean u tx + u xx + u xxxx + (uu x ) x = 0. (1.5) The Camassa-Holm equation have important applications in different physical situations such as waves on shallow water, and processes in semiconductors with differential conductivity [BS76,FPS01,Ros89,Shu87,SG69,Zh05]. This paper 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 [MP98,MP01,MP04]; more precisely, on the choice of test functions according to initial and boundary conditions under consideration.
Here, we give a simple case of the analysis of a rough blow-up, i.e., the case where the solution tends to infinity as t → T * on [0, L]; more exactly, when the integral tends to infinity as t → T * for the given function φ.
Recently, the study of blowing-up solutions of time-fractional nonlinear partial differential equations received great attention. For example, the authors of this paper obtained results on the blow-up of the solutions of time-fractional Burgers equation [AKT19a,Tor19] and fractional reaction-diffusion equation [AKT19b]. We note that the blow-up of the solution of various nonlinear fractional problems was investigated in [AAAKT15, AAKMA17, CSWSS18, KNS08, Pav18, XX18].
Let us briefly describe the problems investigated in this paper: • Blowing-up solutions of the time-fractional Rosenau-KdV-BBM-Burgers equation with initial conditions described as follows: where a, b, c, d ∈ R and u 0 is a given function.
• Blowing-up solutions of the initial-boundary problem for the time-fractional Camassa-Holm-Degasperis-Procesi equation where a, b, c, d ∈ R and u 0 is a given function.
• Blowing-up solutions of the time-fractional Ostrovsky equation with initial conditions: where a, b ∈ R and u 0 is a given function.
• Blowing-up solutions of the initial problem for the time-fractional analogue of the modified Korteweg-de Vries-Burgers equation with dissipation: where a, b ∈ R and u 0 is a sufficiently smooth function.
• Maximum principle and gradient blow-up in time-fractional Burgers equation where ν > 0 and u 0 is a sufficiently smooth function.
The Riemann-Liouville fractional derivative D α +a of order α > 0 (m − 1 < α < m, m ∈ N) is defined as (1.15) Finite time blow-up of solutions of a fractional differential equation. We consider the fractional differential equation (1.16) The blow-up of solutions to (1.16) is assured by the following theorem.
Theorem 1.6. [HKL14] If u 0 > 0, then the solution of problem (1.16) blows-up in a finite time

Blowing-up solutions of the time-fractional Rosenau-KdV-BBM-Burgers equation
In this section we consider the time-fractional Rosenau-KdV-BBM-Burgers equation: where a, b, c, d ∈ R and u 0 is a given function. We study the question of the blow-up of a classical solution u ∈ C 1,4 Let us consider a function ϕ ∈ C 4 ([0, L]) and suppose that the solution u ∈ Multiplying equation (2.1) by ϕ and integrating by parts, we obtain Let the function ϕ(x) be monotonically nondecreasing: and satisfy the following properties (2.4) Then we have Using the Hölder inequality, we obtain the following estimate Then, expression (2.2) takes the form Then the following theorem holds.
Since the functionF (t) is an upper solution of equation Note that the trial function method has great practical convenience.

Blowing-up solutions of the time-fractional Camassa-Holm-Degasperis-Procesi equation
In this section we consider the time-fractional Camassa-Holm-Degasperis-Procesi equation: Multiplying equation (3.1) by ϕ and integrating by parts, we obtain Let 3d − c ≥ 0 and the function ϕ(x) be monotonically nondecreasing: (3.5) Let ϕ satisfy the following properties (3.6) Then we have .
Using the Hölder inequality, we obtain the following estimate
The Theorem 3.1 can be proved as Theorem 2.1. Below we give some examples.

Blowing-up solutions of the time-fractional Ostrovsky equation
We consider the equation with Cauchy data where a, b ∈ R and u 0 is a sufficiently smooth function.
The Theorem 4.1 can be proved as Theorem 2.1.

Blowing-up solutions of the time-fractional modified KdV-Burgers equation
Consider the initial value problem for the time-fractional analogue of the wellknown modified Korteweg-de Vries-Burgers equation with dissipation: 2) where a, b ∈ R and u 0 is a sufficiently smooth function.
Then the following theorem holds.
Example 5.2. Consider problem (5.1) with a = 2 and b = 3 on the interval [0, 1], supplemented with Dirichlet type boundary conditions The purpose of this section is to study time-fractional Burgers equation with the initial condition where ν > 0 and u 0 is a sufficiently smooth function.
6.1. Maximum principle. In this subsection, we present a maximum principle for the time-fractional Burgers equation (6.1).
Then the following theorem holds.
We do not provide the proof this theorem as it runs parallel to that of Theorem 5.1.

Conclusion
In this article, we have studied blowing-up solutions to some time-fractional nonlinear partial differential equations. In precise terms, we have obtained the following results: •