Blowup in $L^1(\Omega)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $\Omega$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(\Omega)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

This article is concerned with the following initial-boundary value problem for a nonlinear time-fractional diffusion equation: where p > 0 is a constant.The left-hand side of the time-fractional differential equation in equation (1.1) means that u(x, • ) − a(x) ∈ H α (0, T ) for almost all x ∈ Ω.For 1 2 < α < 1, since v ∈ H α (0, T ) implies v(0) = 0 by the trace theorem, we can understand that the lefthand side means that u(x, 0) = a(x) in the trace sense with respect to t.As a result, this corresponds to the initial condition for α > 1  2 , whereas we do not need any initial conditions for α < 1  2 .There are other formulations for initial-boundary value problems for time-fractional partial differential equations (e.g., Sakamoto and Yamamoto [25] and Zacher [32]), but here we do not provide comprehensive references.In the case of α = 1, concerning the non-existence of global solutions in time, there have been enormous works since Fujita [9], and we can refer to a comprehensive monograph by Quittner and Souplet [24].We can refer to Fujishima and Ishige [8] and Ishige and Yagisita [13] as related results to our first main result Theorem 1 stated below.See also Chen and Tang [4], Du [6], Feng et al. [7], and Tian and Xiang [29].
Our approach is based on the comparison of solutions to initial value problems for timefractional ordinary differential equations, which is similar to that by Ahmad et al. [2] in the sense that the scalar product of the solution with the first eigenfunction of the Laplacian with the boundary condition is considered.Vergara and Zacher, in their study [30], discuss stability, instability, and blowup for time-fractional diffusion equations with super-linear convex semilinear terms.
To the best knowledge of the authors, there are no publications providing an upper bound of the blowup time for the time-fractional diffusion equation in L 1 (Ω)-norm, which is weaker than L q (Ω)-norm with 1 < q ≤ ∞.
Throughout this article, we assume 3 4 < γ ≤ 1.First, for p > 1, we recall a basic result on the unique existence of local solutions in time.For a ∈ H 2γ (Ω) satisfying ∂ ν a = 0 on ∂Ω and a ≥ 0 in Ω, Luchko and Yamamoto [20] proved the unique existence, which is local in time t.There exists a constant T > 0 depending on a such that (1.1) possesses a unique solution u such that and u ≥ 0 in Ω × (0, T ).The time length T of the existence of u does not depend on the choice of initial values and only depends on a bound m 0 > 0 such that a H 2γ (Ω) ≤ m 0 , provided that ∂ ν a = 0 on ∂Ω.We call T α,p,a > 0 the blowup time in L 1 (Ω) of the solution to (1.1) if As the non-existence of global solutions in time, in this article we are concerned with the blowup in L 1 (Ω).Now we are ready to state our first main results on the blowup with an upper bound of the blowup time for p > 1.
Theorem 1.Let p > 1 and a ∈ H 2γ (Ω) satisfy ∂ ν a = 0 on ∂Ω and a ≥ 0, ≡ 0 in Ω.Then, there exists some T = T α,p,a > 0 such that the solution satisfying (1.2) exists for 0 < t < T α,p,a and (1.3) holds.Moreover, we can bound T α,p,a from above as: We note that T * (α, p, a) decreases as Ω a(x) dx increases for arbitrarily fixed p and a.Meanwhile, T * (α, p, a) tends to ∞ as p > 1 approaches 1, which is consistent because p = 1 is a linear case and we have no blowup.
(2) Estimate (1.4) corresponds to the estimate in [24, Remark 17.2(i) (p.105)] for α = 1.On the other hand, in the case of parabolic equations ∂ t u = D△u + u p with constant D > 0, Ishige and Yagisita discussed the asymptotics of the blowup time T p,a (D) and established ).The principal term of the asymptotics coincides with the value obtained by substituting α = 1 in T * (α, p, a) given by (1.4).Thus, T * (α, p, a) is not only one possible upper bound of the blowup time for 0 < α < 1 but also seems to capture some essence.Moreover, Ishige and Yagisita [13] clarifies the blowup set; see also the work of Fujishima and Ishige [8].For 0 < α < 1, there are no such detailed available results.
The second main result is the global existence of solutions to (1.1) for 0 < p < 1.
In Theorem 2, we cannot further conclude the uniqueness of the solution.This is similar to the case of α = 1, where the uniqueness relies essentially on the Lipschitz continuity of the semilinear term u p in u ≥ 0. Indeed, we can easily give a counterexample by a time-fractional ordinary differential equation: where y ∈ H α (0, T ).Then, we can directly verify that both y(t) = t 2α and y(t) ≡ 0 are solutions to this initial value problem.
The key to the proof of Theorem 1 is a comparison principle [20] and a reduction to a time-fractional ordinary differential equation.Such a reduction method can be found in the studies by Kaplan [14] and Payne [21] for the case α = 1.On the other hand, Theorem 2 is proved by the Schauder fixed-point theorem with regularity properties of solutions [31].For a related method for Theorem 2, we refer to the study by Díaz et al. [5].
This article is composed of five sections; in Section 2, we show lemmata that complete the proof of Theorem 1 in Section 3; we prove Theorem 2 in Section 4; finally, Section 5 is devoted to concluding remarks and discussions.

Preliminaries
We will prove the following two lemmata.

Completion of proof of Theorem 1
Step 1.We set where a 0 := Ω a(x) dx.Here, we see that a 0 > 0 because a ≥ 0, ≡ 0 in Ω by the assumption of Theorem 1.
Remark 2. We note that η(t) is the inner product of the solution u( • , t) with the first eigenfunction 1 of −△ with the homogeneous Neumann boundary condition.As for the parabolic case, we can refer to the studies by Kaplan [14] and Payne [21].
Henceforth, we assume that the solution u to (1.1) within the class (1.2) exists for 0 < t < T .By (1.2), we have Ω (u(x, t) − a(x)) dx ∈ H α (0, T ).Fixing ε > 0 arbitrarily small, we see and hence, Since ∂ ν u = 0 on ∂Ω × (0, T − ε), Green's formula and the governing equation On the other hand, introducing the Hölder conjugate q > 1 of p > 1, i.e., 1 q + 1 p = 1, it follows from u ≥ 0 in Ω × (0, T − ε) and the Hölder inequality that By (3.1) and (3.2), we obtain Step 2. This step is devoted to the construction of a lower solution η(t) satisfying We restrict the candidates of such a lower solution to and thus, d dt Next, by termwise differentiation, we have for 0 ≤ t ≤ T − ε.Repeating the calculations and by induction, we reach Plugging (3.7) into (3.6),we obtain d dt Then, by the definition of d α t , we calculate Here, we employ integration by substitution s = tξ and the beta function to treat Then, we can bound d α t ( 1 (T −t) m ) from above as follows: For the series above, we utilize (3.6) and (3.7) again to find Recalling the definition (3.5) of η(t), we eventually arrive at (3.8) Note that (3.8) holds for arbitrary m ∈ N, T > 0, and 0 < t < T − ε.
Finally, we claim that for any p > 1 and a 0 > 0, there exist constants m ∈ N and T > 0 such that (3.9) In fact, (3.9) is achieved by Since ε > 0 was arbitrarily chosen, we obtain Since η(t) = u( • , t) L 1 (Ω) , this means that the solution u cannot exist for t > T * (α, p, a).Hence, the blowup time T p,a ≤ T * (α, p, a).The proof of Theorem 1 is complete.

Proof of Theorem 2
Step 1. Henceforth, we denote the norm and the inner product of L 2 (Ω) by respectively.We show the following lemma.
Proof.By 0 < p < 1, we see that 1 1−p > 1, and the Hölder inequality yields which completes the proof for w.The proof for η is the same.
Let A = −△ with D(A) = {w ∈ H 2 (Ω); ∂ ν w = 0 on ∂Ω}.We number all the eigenvalues of A as 0 with their multiplicities.By {ϕ n } n∈N , we denote the complete orthonormal basis of L 2 (Ω) formed by the eigenfunctions of A, i.e., Aϕ n = λ n ϕ n and ϕ n = 1 for n ∈ N. We can define the fractional power A β for β ≥ 0, and we know that a H 2β (Ω) ≤ C A β a for all a ∈ D(A β ), where the constant C > 0 depends on β, Ω (e.g., [18,22]).
Lemma 4(i) implies S(t)a ≤ C a , t ≥ 0. (4.1)We choose a constant M > 0 sufficiently large such that Since 0 < p < 1, we can easily verify the existence of such M > 0 satisfying (4.2).
With this M > 0, we define a set B ⊂ L 2 (0, T ; L 2 (Ω)) by We define a mapping L by
Consequently, it is verified that the fixed-point u satisfies (1.2).By (1.2) and (4.10) we see that u satisfies (1.1) in terms of [31,Lemma 5].Thus, the proof of Theorem 2 is complete.
As g(t), we take a similar function to (3.5): Then, by (3.8) we have Therefore, for mp − (m + 1) ≥ 0, it suffices to choose T > 0 such that ≤ ξ mp−(m+1) for all ξ ≥ 1 by setting ξ := T T −t ≥ 1.Hence, g(t) is a lower solution if for mp ≥ m + 1. Choosing the minimum m ∈ N and arguing similarly to the final part of the proof of Theorem 1, we obtain an inequality for the blowup time T α,p,a (∞) in L ∞ (Ω): where [q] denotes the maximum natural number not exceeding q > 0.

2.
Restricting the nonlinearity to the polynomial type u p , in this article, we investigate semilinear time-fractional diffusion equations with the homogeneous Neumann boundary condition.With nonnegative initial values, we obtained the blowup of solutions with p > 1 as well as the global-in-time existence of solutions with 0 < p < 1.The key ingredient for the latter is the Schauder fixed-point theorem, whereas that for the former turns out to be a comparison principle for time-fractional ordinary differential equations (see Lemma 2) and the construction of a lower solution of the form (3.5).We can similarly discuss the blowup for certain semilinear terms like the exponential type e u and some coupled systems.More generally, it appears plausible to consider a general convex semilinear term f (u), which deserves further investigation.
Technically, by introducing we reduce the blowup problem to the discussion of a time-fractional ordinary differential equation.As was mentioned in Remark 2, indeed 1 is the eigenfunction for the smallest eigenvalue 0 of −△ with ∂ ν u = 0. On this direction, it is not difficult to replace −△ with a more general elliptic operator.Actually, in place of 1, one can choose an eigenfunction ϕ 1 for the smallest eigenvalue λ 1 and consider η(t) := (u( • , t), ϕ 1 ) L 2 (Ω) to follow the above arguments.In this case, it is essential that λ 1 ≥ 0 and ϕ 1 does not change sign.We can similarly discuss the homogeneous Dirichlet boundary condition.

3.
In the proof of Theorem 1, we obtained an upper bound T * (α, p, a) of the blowup time T (see (1.4)), but there is no guarantee for its sharpness.Sharp estimates for the blowup time in the time-fractional case is expected to be more complicated than the parabolic case, which is postponed to a future topic.
We briefly investigate the monotonicity of In particular, f (α) cannot be monotone increasing for C p,a < 1 and cannot be monotone decreasing for C p,a > 1, which implies C * ≥ 1 and C * ≤ 1.
4. Related to the blowup, we should study the following issues: (i) Lower bounds or characterization of the blowup times.
(ii) Asymptotic behavior or lower bound of a solution near the blowup time.
(iii) Blowup set of a solution u(x, t), which means the set of x ∈ Ω, where |u(x, t)| tends to ∞ as t approaches the blowup time.For α = 1, comprehensive and substantial works have been accomplished.We are here restricted to refer to Chapter II of Quittner and Souplet [24] and the references therein.However, for 0 < α < 1, by the memory effect of ∂ α t u( • , t) which involves the past value of u, several useful properties for discussing the above issues (i)-(iii) do not hold.Thus, the available results related to the blowup are still limited for 0 < α < 1, and it is up to future studies to pursue (i)-(iii).