Blow-Up Solutions for the Space-Time Fractional Evolution Equation

This paper focuses on the blow-up solutions of the space-time fractional equations with Riemann–Liouville type nonlinearity in arbitrary-dimensional space. Using the Banach mapping principle and the test function method, we establish the local well-posedness and overcome the difficulties caused by the fractional operators to obtain the blow-up results. Furthermore, we get the precise lifespan of blow-up solutions under special initial conditions.


Introduction
In this paper, we consider the following problem for the space-time fractional evolution equation: where ∈ (0, 1) , ∈ (0, 2] , p > 1 , 0 (x) ∈ C 0 ℝ N .The nonlocal fractional Lapla- cian operator (−Δ) ∕2 is realized as a Fourier multiplier with symbol | | : under Fourier transform, D 0|t is the Caputo fractional deriva- tive operator, and I 1− 0|t denotes the Riemann-Liouville (R-L) fractional integral operator; they are defined respectively as (1) where Γ is the Euler gamma function.C 0 ℝ N is the normal space in which all con- tinuous functions decay to zero as they approach infinity.
For this, Nagasawa [11], Kobayashi [12], Guedda and Kirane [13] considered the evolution equation involving fractional diffusion Additionally, Cazenave, Dickstein and Weissler [14] proved the sharp blow-up, global existence results for the heat equation with nonlinear memory, Fino and Kirane [15] generalized and solved the associated problems of the following equations based on article [14]: Using an existence-uniqueness test, they confirmed the validity of the equation and proved the existence of a blow-up solution.When the solution to problem (4) is blow-up in finite time.In addition, the conditions for local or global solutions have also been established.
Different from previous work, in this paper, we focus on the evolution equations with two fractional forms of the Caputo time and the fractional diffusion.Our interest in this problem is motivated by the studies above to develop a general blow-up theory for (1)with R-L type nonlinear term.For this, we perform a fixed point argument to establish the local well-posedness.Our proof relies on the properties of the space-time fractional operators derived from Mittag-Leffler functions [19].In addition, by contradiction argument, we obtain sufficient conditions for blow-up.Usually we need to define the mild solution as follows: Journal of Nonlinear Mathematical Physics (2023) 30:917-931 then is a mild solution to the problem (1).For the specific definition of V , , K , , see the preliminaries.
However, due to the lack of space-time estimates for V , , K , , we cannot directly derive the blow-up of the solutions.In order to overcome the technical difficulty, we introduce an integral test function which allows us to deal with the nonlinearity properly, and then use the relation of the R-L type operators and the Mittag-Leffler operators to show the equivalence between mild solutions and weak solutions.It turns out that weak solutions work well for achieving our goal and obtaining the threshold of p.It is worth noticing that this threshold will tend to the one obtained in [15] as → 1 .Moreover, we shall present the upper bound of the lifespan of solu- tions for some special initial data, the proof of this point is standard.
For simplicity, we use f ≲ g to denote f ≤ Cg , where C may have different val- ues in different lines.Our main results are summarized below.
) is a mild solution of the problem (1), then is also a weak solution.
holds, the lifespan T of the solution (x, t) of the problem (1) has the following upper bound: is holding for some positive constant 0 .Then, for any ∈ [ 0 , +∞) , the lifespan T of the solution (x, t) to the problem (1) satisfies the following bound: The remainder of this paper is divided into two parts.In part 2, we collect the necessary definitions and Lemmas.The part 3 is devoted to the proof of our main conclusions.

Preliminaries
In this section, we outline and review the relevant properties about the evolution operators, which are essential to prove our main conclusions.
Given Z(t) ∶= e −t(−Δ) ∕2 , where (−Δ) ∕2 is a self-adjoint operator on L 2 (ℝ N ) , it follows that Z(t) is a strongly continuous semigroup on L 2 (ℝ N ) generated by (−(−Δ) ∕2 ) (see [16]).Z(t) = J (t) * , where * stands for convolution, and Using the self-similar form of J (x, t) and convolution by Young's inequality, we can conclude Journal of Nonlinear Mathematical Physics (2023) 30:917-931 Regarding the representation (5) of the mild solution, it contains the relevant content of the Mainardi's and Mittag-Leffler functions.In the following, we will present their definitions.
A function of the Mittag-Leffler form with two parameters is defined as The semigroup Z(t) whose Mittag-Leffer operators forms are defined as follows: and where M( ; ) is a Mainardi's function defined by M( ; ) ≥ 0 for all ≥ 0 and satisfies the following properties: From this we can derive the following results.

◻
In addition, we review the previous conclusions concerning time fractional operators.

Lemma 2.2 ([17])
Next, let us define R-L fractional derivatives and recall several results that will be used in proving the equivalence of the mild solution and the weak one.Definition 2.4 Let T > 0 , ∈ AC[0, T] , AC stands for the space of absolutely con- tinuous functions.Fractional derivatives of order ∈ (0, 1) on the left-and right- sides of the R-L are defined as follows: and Based on this definition, it is not difficult to derive the following relation between Caputo and R-L derivatives
Similarly, we can complete the proof of another equation.◻

Proof of Main Results
Proof of Theorem 1. 2 We prove this conclusion based on the contraction mapping principle.A Banach space Π T is constructed for every T > 0, . For any given ∈ Π T , we define ) by using ( 11) and ( 12), then By choosing T small enough, we have this implies that ‖G( )‖ 1 ≤ 2‖ 0 ‖ ∞ .Consequently, we get G( ) ∈ Π T .∶ For , ∈ Π T , by using (12), we have the following estimate, due to the inequality a choice of small T such that Journal of Nonlinear Mathematical Physics (2023) 30:917-931 implies that G( ) is a contraction mapping on Π T .To sum up, we conclude from Banach's fixed point theorem that there is a mild solution to the problem (1).∶ Concerning the uniqueness issue, we use Gronwall's inequality to deal with it.
Let 1 , 2 be two mild solutions in Π T .Using ( 12) and ( 16), we obtain From Gronwall's inequality, we infer that 1 = 2 .In addition, due to the unique- ness, there must be a solution in the maximal interval [0, T max ) (see also Fino, Kirane [15]).If 0 ≥ 0 and 0 ≢ 0 , by ( 9), (10), we can get directly from ( 5) that  ≥ V , (t) 0 > 0 .This closes the proof.◻ Proof of Theorem 1.4 Equation ( 5) implies that By Lemma 2.2, we get Integrating the above equation with respect to the variable x yields Now, using Lemma 2.3, one has Next, we construct the time derivative of M 2 , let h > 0, t + h ≤ T , and obtain By dominated convergence theorem, we deduce that when h tends to zero, J 1 and J 3 respectively converge to Afterwards, we consider the estimation of J 2 , which can be rewritten as follows: By dominated convergence theorem, we deduce that when h tends to zero, J 2 con- verge to Then, we get 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:917-931 Using the integration by parts formula, we derive

It follows
The proof is completed.◻ Proof of Theorem 1. 5 Here, we use the contradiction analysis based on the test functions to verify our conclusion.In what follows, we prove this conclusion in two different cases associated with p > 1.

It has
where we used the estimate f −1 (−Δ) ∕2 f ≥ (−Δ) ∕2 f for any bounded and con- tinuous function f ≥ 0 and all ≥ 1 [20].Applying Young's inequality with 1 p + 1 q = 1, p, q > 1 , and taking the weight coefficient = 1 4p , we consequently get that Similarly, taking Q 2 = [0, 1] × {y ∈ ℝ N ;|y| ≤ 2} , and substituting = t T , y = x T ∕ , (−Δ x ) 2 = T − (−Δ y ) 2 , we have the remaining two terms of the integral are still bounded in Q 2 .Let- ting T → ∞ we can obtain that the right terminal term of ( 20) is zero, while the left terminal term is positive.Therefore, we obtain a contradiction when T → ∞ .While Taking T sufficiently large, we obtain a similar result.
∶ The proof for the case p < 1∕ is similar to the case 1 < p < 1 +  N , we redefine the test function

3
Journal of Nonlinear Mathematical Physics (2023) 30:917-931 , R ∈ (0, T) , T and R cannot be infinite at the same time.The defini- tion of Ψ is the same as in Case 1.