The ﬁnite time blow-up for Caputo-Hadamard fractional di ﬀ usion equation involving nonlinear memory

: In this article, we focus on the blow-up problem of solution to Caputo-Hadamard fractional di ﬀ usion equation with fractional Laplacian and nonlinear memory. By virtue of the fundamental solutions of the corresponding linear and nonhomogeneous equation, we introduce a mild solution of the given equation and prove the existence and uniqueness of local solution. Next, the concept of a weak solution is presented by the test function and the mild solution is demonstrated to be a weak solution. Finally, based on the contraction mapping principle, the ﬁnite time blow-up and global solution for the considered equation are shown and the Fujita critical exponent is determined. The ﬁnite time blow-up of solution is also conﬁrmed by the results of numerical experiment.

We next recall some pioneering work on the blow-up problem for fractional diffusion equation, here we only mention the results related to our studies.
Recently, Li and Li [16] investigated the semilinear time-space fractional diffusion equation involving Caputo-Hadamard derivative and fractional Laplacian, where 0 < α < 1, 0 < s < 1, p > 1, and u a ∈ C 0 (R d ). They obtained that: If 1 < p < 2s d and u a ≥ 0 with u a 0, then the solution of (1.6) will blow up in finite time; Conversely, if p ≥ 1 + 2s d and ||u a || L q * (R d ) is sufficiently small, where q * = d(p−1) 2s , then (1.6) has a global solution. Motivated mathematically by the results and methods in [16], this paper will further study the blow-up property and global solution to time-space fractional diffusion equation (1.1) with nonlinear memory. The main result is displayed in the following theorem.
αd }, then the mild solution of Eq (1.1) will blow up in finite time.
The organization of this paper is as follows. Section 2 recalls some basic definitions and presents several important lemmas. In Section 3, we define a mild solution to Eq (1.1) and then prove the local existence and uniqueness of the mild solution. Then, a weak solution of Eq (1.1) is introduced and the mild solution is actually proved to be a weak solution. Next, we show the finite time blow-up and global existence of the solution to Eq (1.1) in Section 4. Finally, an illustrative example is provided to verify the blow-up of solution in finite time in Section 5. The conclusions are given in the last section. Throughout the paper, we use the letter C to denote a generic positive constant which may take different values at different places.

Preliminaries
Let us recall some basic definitions and several important lemmas, which will be applied in the next sections.
Definition 2.1. [4,30] Let a function f (t) be defined on the interval (a, b) (0 ≤ a < b ≤ +∞) and α > 0. The left-and right-sided Hadamard fractional integrals of the function f (t) with order α are given by
To define a mild solution of Eq (1.1), let us consider the following linear equation, whose solution is expressed by [14] u( where G a (x, t) and G f (x, t) are the fundamental solutions given by . (2.9) The special function H 21 23 (z) in the above equalities is the Fox H-function and some details regarding this function can be found in [4,32,33].
In the sequel, we list some properties of the functions G a (x, t) and G f (x, t).
And there exists a constant C > 0 such that For simplicity of representation, from now on, we denote , and so on.

The mild solution and weak solution
In this part, we first define a mild solution of Eq (1.1) and then prove the local existence and uniqueness of the mild solution in terms of the contraction mapping principle. Next, the definition of a weak solution is introduced to Eq (1.1). We can also prove that the mild solution is just a weak solution. Let us begin by introducing the definition of a mild solution to Eq (1.1).
Obviously, (E a,T , d) is a complete metric space. By means of the fundamental solutions G a (t) and G f (t), we define the following operator F on the metric space (E a,T , d), It follows from Lemma 2.5 that F (u) ∈ C([a, T ], C 0 (R d )). We next show that F : E a,T → E a,T . For u ∈ E a,T and t ∈ [a, T ], by Definition 2.1 and Lemma 2.1, we get Choosing T > a sufficiently close to a such that We need to show that the operator F is contractive on E a,T . For u, v ∈ E a,T and t ∈ [a, T ], one can deduce that Taking T > a sufficiently close to a gives rise to . This illustrates the operator F is contractive on E a,T and thus it has a fixed point u ∈ E a,T by the contraction mapping principle. Moreover, using Gronwall inequality immediately knows the uniqueness of the mild solutions to Eq (1.1) holds.
In view of the uniqueness, there is a maximal time T max > a such that the solution of Eq (1.1) exists on the interval [a, T max ), where . Furthermore, using For h > 0 and σ > 0, consider a set Then the metric space ( E h,σ , d) is complete. On the space ( E h,σ , d), define an operator Q as follows, It is easy to see that Q By taking sufficiently small h, we arrive at In regard to ||J 2 || L ∞ (R d ) , one has In this case, for t ∈ [T max , T max + h], one may take very small h such that which suggests the operator Q is contractive on E h,σ and thus it has a fixed point v ∈ E h,σ . In view of v(T max ) = Q(v)(T max ) = u(T max ), we set such that u(t) ∈ C([a, T max + h], C 0 (R d )) and which means u(t) is indeed a mild solution of Eq (1.1). Recalling the definition of T max , this yields a contradiction. The proof of the remainder of this theorem follows that of Theorem 3.2 in [16] and so is omitted. The proof is thus complete.
In the following, we present the definition of a weak solution to Eq (1.1) and show that the mild solution given by Definition 3.1 is a weak solution.
Proof. Assume that u ∈ C([a, T ], C 0 (R d )) is a mild solution to Eq (1.1). Then Definition 3.1 gives

Use Lemma 2.5 to get
Therefore, for every ϕ ∈ C 2,1 x,t (R d × [a, T ]) satisfying supp x ϕ ⊂⊂ R d and ϕ(·, T ) = 0, there holds For I 1 , an application of Lemma 2.4 leads to To estimate I 2 , we set h > 0, t ∈ [a, T ) and t + h ≤ T , then Applying the mean value theorem yields that Consequently, which is the desired result and the proof is now ended.

Proof of main result
Proof of Theorem 1.1.
(1) We consider two cases: Thanks to Theorem 3.2, we may take . From Definition 3.2 of the weak solution, one has According to the inequality (−∆) s ω(x) ≤ ω(x) in [16] and the Lebesgue dominated convergence theorem, we have with n → ∞ in (4.2), Using Jensen's inequality in (4.3) gives Denoting f (t) = R d u ω dx, it is easy to see that f (t) ≥ 0 and f (a) > 0. In view of inequality (4.4), Hölder inequality and Young's inequality, we obtain Hence there holds Then we get If Eq (1.1) has a global solution, we know that f (a) = 0 as T → ∞ in (4.5) by 0 < α < γ < 1 and p < 1 + 1−γ α , which is inconsistent with f (a) > 0. Hence, the mild solution of Eq (1.1) blows up in finite time.
According to (4.6)-(4.8), together with Young's inequality and Hölder inequality, it holds that As a result, i.e., (4.11) The condition 1 < p < 1+ 2s(1+α−γ) then R d u a ϕdx = 0 as T → ∞, that is u a ≡ 0, which makes a contradiction with the assumption u a 0. Therefore, blowup of the mild solution u of Eq (1.1) occurs in finite time.

(4.22)
Choose sufficiently small ϑ and K such that This implies Ψ(u) ∈ E K and thus Ψ has a fixed point u ∈ E K by the contractive mapping principle. Finally, We need to prove u ∈ C([a, ∞), C 0 (R d )). For a T sufficiently close to a, let As demonstrated before, it is known that there is a unique solution u on E K,T . It follows from Theorem 3.1 and the initial value u a ∈ C 0 (R d ) ∩ L q (R d ) that there exists a unique solution u ∈ C([a, T ], C 0 (R d )) ∩ C([a, T ], L q (R d )) for T sufficiently close to a. Hence, for T sufficiently close to a, one has sup a<t<T log t a β || u(t)|| L q (R d ) ≤ K. This means that u = u for t ∈ [a, T ] from the uniqueness of solution and thus u ∈ C([a, T ], C 0 (R d )) ∩ C([a, T ], L q (R d )).
Our purpose is to prove u ∈ C([a, ∞), C 0 (R d )). In fact, for t > T , it holds that Using the fact u ∈ C([a, T ], C 0 (R d )), one obtains For any T > T , it can be easily find that u p ∈ L ∞ ((T, T ), L q/p (R d )) and H D −(1−γ) a,τ u p ∈ L ∞ ((T, T ), L q/p (R d )). On the other hand, the condition q > d(p−1) 2s indicates that we may choose r > q such that d 2s ( p q − 1 r ) < 1. As what we have proved in Lemma 2.5, it is obvious that I 2 ∈ C([T, T ], L r (R d )). By the arbitrariness of T , we see that I 2 ∈ C([T, ∞), L r (R d )) and thus u ∈ C([T, ∞), L r (R d )).
Remark 4.1. It is worth noticing that, according to Theorem 1.1, the Fujita critical exponent to Eq (1.1) is the number p = max{1 Remark 4.2. In the Eq (1.1), we consider the case 0 < α < γ < 1 and prove the main result, i.e., Theorem 1.1. If γ ≥ α with 0 < α < 1 and 0 ≤ γ < 1, then it is easy to verify that Theorems 3.1 and 3.2 are still valid provided that a mild solution and a weak solution are defined as Definitions 3.1 and 3.2. However, compared with Theorem 1.1, we see that the main conclusions are very different. In fact, we can derive the following result whose proof is similar to that of Theorem 1.1 or can also refer to the proof of Theorem 1 in [34].

Numerical simulations
In this section, we show the finite time blow-up of the solution to Eq (1.1) by numerical simulation. For this purpose, we have to approximate the Caputo-Hadamard derivative, fractional Laplacian and Hadamard fractional integral in Eq (1.1), respectively. We shall use formulaes (3.2) and (3.3) in [35] to discretize the Caputo-Hadamard derivative of order α ∈ (0, 1) and apply formula (2.9) in [36] to approximate the fractional Laplacian of order s ∈ (0, 1). For the right sided Hadamard fractional integral of order 1 − γ (γ ∈ (0, 1)) in Eq (1.1), we present the following discrete scheme.
Let a = t 0 < t 1 < . . . < t k < . . . < t N = T be a partition of the interval [a, T ] with N ∈ N and some positive number T > a. Then the Hadamard fractional integral with order 1 − γ (γ ∈ (0, 1)) can be approximated by, for t = t k , 1 ≤ k ≤ N, Based on these results, we obtain a numerical scheme to Eq (1.1). For simplicity, we now take d = 1, a = 1, p = 2 and u a = 10 in Eq (1.1). Figure 1 depicts the curves of the solution to Eq (1.1) when the parameters α and s choose different values and γ = 0.8, which displays the finite time blowup of solution of Eq (1.1) and thus shows the effectiveness of the results in Theorem 1.1. Similarly, Figure 2 presents the curves of the solution to Eq (1.1) in the case γ ≤ α and illustrates the validity of the results given by Theorem 4.1.

Conclusions
In this paper, we study the blow-up and global existence of solution of the Cauchy problem to time-space fractional partial differential Eq (1.1) with nonlinear memory. A mild solution and a weak solution are introduced to Eq (1.1) and the mild solution is actually shown to be the weak solution. We next prove the local existence and uniqueness of the mild solution of Eq (1.1) by using the fixed point argument. Finally, the finite time blow-up and global solution of Eq (1.1) are established and the Fujita critical exponent is also determined, where the blowing-up character of the solution in a finite time is verified by numerical simulations.