Life Span of Solutions for a Time Fractional Reaction-Diffusion Equation with Non-Decaying Initial Data

Takefumi Igarashi

I. INTRODUCTION

We study the Cauchy problem for a time fractional reaction-diffusion equation

where n ≥ 1, 0 < α < 1, p > 1, u0 ∈ C(Rn) ∩ L∞(Rn), where $n \geq 1$ , $0 < \alpha < 1$ , $p > 1$ , $u_0 \in C(\mathbf{R}^n) \cap L^\infty(\mathbf{R}^n)$ , and $\partial_t^\alpha$ denotes the Caputo time fractional derivative of order $\alpha$ defined by

Here, $\Gamma(\cdot)$ is the Gamma function. Moreover, the Caputo time fractional derivative (1.2) is related to the Riemann-Liouville derivative by

In this paper, we show that every solution of (1.1) blows up in finite time with the non-decaying initial data at space infinity, and also present the estimate on the life span of the solutions for (1.1). Then, we define the life span (or blow-up time) $T^{*}$ as

$T^{*} = \sup \{T > 0$ ; there exists a mild solution $u$ of (1.1) in $C([0,T],C(\mathbf{R}^n)\cap L^\infty (\mathbf{R}^n))\}$ , where the definition of "mild solution" and the local "existence" of a mild solution are described in section 2. If $T^{*} = \infty$ , the solution is global. On the other hand, if $T^{*} < \infty$ , then the solution is not global in time in the sense that it blows up at $t = T^{*}$ such as

A brief review on the blow-up and global existence results obtained for Cauchy problem (1.1) is given below:

(A) Kirane et al. [12] obtained that the following results.

If $1 < p \leq 1 + 2\alpha / \{\alpha n + 2(1 - \alpha)\}$ , then (1.1) admits no global weak nonnegative solutions other than the trivial one.
Let $u$ be a local solution to (1.1). Then, there exists a constant $C > 0$ such that

where $0 < t \leq T < +\infty$

Suppose that (1.1) has a nontrivial global nonnegative weak solution. Then, there is a constant $K > 0$ such that

(B) When $u_0 \in C_0(\mathbf{R}^n):= \{u \in C(\mathbf{R}^n) | \lim_{|x| \to \infty} u(x) = 0\}$ , the following results were proved by Zhang and Sun [28] and Zhang et al. [29]:

If $1 < p < 1 + 2/n$ , then any nontrivial positive solution of (1.1) blows up in finite time.
If $p \geq 1 + 2 / n$ and $\| u_0\|_{L^{qc}}$ is sufficiently small, where $q_{c} = n(p - 1) / 2$ , then (1.1) has a global solution.
If $\int_{\mathbf{R}^n}u_0(x)\chi (x)dx > 1$ , where

then the solutions of (1.1) blow up in finite time.

(C) The following results were also obtained in Ahmad et.al. [1] when $u_0 \in C_0(\mathbf{R}^n)$ :

If $p \geq 1 + 2 / n$ and $\| u_0\|_{L^1} + \| u_0\|_{L^\infty} \leq \epsilon_0$ with some $\epsilon_0 > 0$ , there exists $s > p$ such that (1.1) admits a global solution with $u \in L^{\infty}([0,\infty),L^{\infty}(\mathbf{R}^{n})) \cap L^{\infty}([0,\infty),L^{s}(\mathbf{R}^{n}))$ . Furthermore, for all $\delta > 0$ ,

In addition, if $pn < 2s$ , or

with

then $u\in L^{\infty}([0,\infty),L^{\infty}(\mathbf{R}^{n}))$

for some constant $\sigma > 0$ .

If $Z_0 \coloneqq \int_{\mathbf{R}^n} u_0(x) \chi(x) dx > 2^{1/(p-1)}$ , then the solutions of (1.1) blow up in finite time, and the estimate of the blow-up time is

Several studies have been made on the life span of solutions. The results are given below:

(A) Gui and Wang [6] and Mukai et al. [17] considered

for $m = 1$ and $m > 1$ , respectively, and proved the following life span results when an initial datum takes the form $v_{0}(x) = \lambda \phi(x)$ , where $\lambda > 0$ and $\phi(x)$ is a bounded continuous in $\mathbf{R}^n$ :

If $\| \phi \|_{L^{\infty}(\mathbf{R}^{n})} = \phi (0) > 0$ , then there exists $\lambda_1\geq 0$ such that $T^{*} < \infty$ for any $\lambda >\lambda_{1}$ , and

If $\| \phi \|_{L^{\infty}(\mathbf{R}^{n})} = \lim_{|x|\to \infty}\phi (x) = \phi_{\infty} > 0$ , then $T^{*} < \infty$ for any $\lambda >0$ , and

(B) Giga and Umeda [4, 5], Seki [19] and Seki et al. [20] showed the solution of (1.4) blows up at minimal blow-up time (see Remark 1 below); that is,

if and only if there exists a sequence $\{x_{j}\} \subset \mathbf{R}^{n}$ such that

Remark 1. Applying the comparison principal to (1.4), it follows that

So, when (1.5) holds, we call the time $T^{*}$ the "minimal blow-up time" and the solution $v$ to (1.4) a "blow-up solution with the minimal blow-up time".

(C) Maingé [15] considered (1.4) for $\max(0,1 - 2/n) < m < 1$ , and proved if the initial data satisfies

where $x_0 \in \mathbf{R}^n$ , $s > 2$ , and $c_0^{p_1 - m} > C_b\phi_0$ for some constant $C_b > 0$ and $\phi_0 > 0$ , then the solution of (1.4) blows up in finite time, and

where $d_1 > 0$ and $d_2 > 0$ .

(D) Yamauchi [18, 25, 26, 27] considered (1.4) for $m = 1$ , the author [8, 9] for $\max(0,1 - 2/n) < m < 1$ or $1 < m < p_1$ , and showed the following life span results:

(a) Let $n\geq 2$ . For some $\xi \in S^{n - 1}$ and $\delta >0$ , we set the conic neighborhood $D_{\xi}(\delta)$

and set $S_{\xi}(\delta) = D_{\xi}(\delta)\cap S^{n - 1}$ . Define

where $r = |x|$ , $\theta = x / r$ .

If $N_{\infty} > 0$ , then the solution of (1.4) blows up in finite time, and

If $N_{\infty} = \| v_0\|_{L^{\infty}(\mathbf{R}^{n})}$ , then the solution of (1.4) blows up at minimal blow-up time; that is,

(b) Let $n = 1$ . Define

If $n_{\infty} > 0$ , then the solution of (1.4) blows up in finite time, and

If $n_{\infty} = \| v_0\|_{L^{\infty}(\mathbf{R})}$ , then the solution of (1.4) blows up at minimal blow-up time; that is,

(E) The author [10] also considered

for $p_2 \geq 1$ or $q \geq 1$ , and showed the following life span results:

(a) Let $n\geq 2$

If $N_{\infty} > 0$ , then the solution of (1.8) blows up in finite time, and

If $N_{\infty} = \| v_0\|_{L^{\infty}(\mathbf{R}^{n})}$ , then the solution of (1.8) blows up at minimal blow-up time; that is,

(b) Let $n = 1$

If $n_{\infty} > 0$ , then the solution of (1.8) blows up in finite time, and

If $n_{\infty} = \| v_0\|_{L^{\infty}(\mathbf{R})}$ , then the solution of (1.8) blows up at minimal blow-up time; that is,

Several recent studies show that the minimal blow-up time is strongly associated with blow-up at space infinity. Related researchers are Giga and Umeda [4, 5], Mochizuki and Suzuki [16], Ozawa and Yamauchi [18], Seki [19], Seki et al. [20], Shimojo [22], Yamaguchi and Yamauchi [27], Yamauchi [25, 26] and the author [8, 9].

Here, we state the main results.

Theorem 1. Consider the Cauchy problem (1.1) for $0 < \alpha < 1$ and $p > 1$ .

(a) Let $n \geq 2$ . Suppose that there exist $\xi \in S^{n-1}$ and $\delta > 0$ such that

where $r = |x|$ , $\theta = x / r$ , $S_{\xi}(\delta) = D_{\xi}(\delta) \cap S^{n-1}$ and $D_{\xi}(\delta)$ is the conic neighborhood defined by (1.7). Then the solution of (1.1) blows up in finite time, and we have

Inparticular,assumingthat

the solution of (1.1) blows up at

(b) Let $n = 1$ . Suppose that

Then the solution of (1.1) blows up in finite time, and we have

In particular, assuming that

the solution of (1.1) blows up at

Theorem 1 allows us the information of the life span for the initial data of intermediate size and the non-decaying initial data at space infinity; (1.9) and (1.13).

Remark 2. We show some examples of the initial data $u_0$ which satisfy $M_{\infty} > 0$ in the space dimensions $n \geq 2$ . For simplicity, we employ polar coordinates.

(i) $u_0(r,\alpha) = 1 - \exp (-r^2)$

(ii) $u_0(r,\alpha) = \{1 - \exp (-r^2)\} (2 - \cos r).$

(iii) $u_0(r,\alpha) = \{1 - \exp (-r^2)\} (1 + \cos \alpha).$

(iv) $u_0(r,\alpha) = \{1 - \exp (-r^2)\} (1 + \cos \alpha)(2 - \cos r).$

For the examples (i) and (iii), the initial data $u_0$ satisfies (1.11). However, for the examples (ii) and (iv), since $\| u_0 \|_{L^\infty(\mathbf{R}^n)} = 3$ and $\| u_0 \|_{L^\infty(\mathbf{R}^n)} = 6$ , respectively, it follows that $M_\infty \neq \| u_0 \|_{L^\infty(\mathbf{R}^n)}$ .

The outline of the rest of this paper is organized as follows. In section 2, we give the existence theorem of a local solution to (1.1). In section 3, we prove the main results by improving the method in the author [8, 9], Ozawa and Yamauchi [18] and Yamauchi [25, 26].

II. EXISTENCE OF A LOCAL MILD SOLUTION

In this section, we show the local existence and uniqueness theorem of a mild solution to problem (1.1). Here, we state the definition of a mild solution of (1.1).

Definition. Let $T^{*} > 0$ . We say $u \in C([0, T^{*}], C(\mathbf{R}^{n}))$ is a mild solution of (1.1) if $u$ satisfies the integral equation

where $f(u(s)) = u^{p}(s)$ , and $A$ is realization of $-\Delta$ and $E_{\alpha, \beta}(z)$ is the Mittag-Leffler function (see [11]):

Theorem 2. Suppose that $u_0 \in C(\mathbf{R}^n) \cap L^\infty(\mathbf{R}^n)$ . Then there exists a unique local mild solution $u \in C([0, T^*], C(\mathbf{R}^n))$ for the problem (1.1).

Proof. See [21, Theorem 1] noting that the nonlinear term $f(u(s)) = u^{p}(s)$ is a locally Lipschizian function. (See also [24, Theorem 2.2].)

Remark 3. If $u$ solves (1.1), then $u$ satisfies (2.1) by the method of the proof for [21, Lemma 1].

Remark 4. If $u_0(x) \geq 0$ , then (1.1) admits a solution which satisfies $u(x,t) \geq 0$ by the maximum principle [2](see also [3, Theorem 3.1] and [21, Theorem 3]).

III. PROOF OF THEOREM 1

In this section, we shall estimate the life span $T^{*}$ both from below and from above. Here, we improve the method in Yamauchi [25, 26] and the author [8, 9, 10].

First, we shall show a lower estimate of $T^{*}$ in the space dimensions $n \geq 1$ . This is obtained by comparing the solution $u$ of (1.1) with the solution $U$ of the ordinary differential equation

The solution $U$ of (3.1) satisfies the integral equation

where $E_{\alpha, \beta}(z)$ is the Mittag-Leffler function by defined in (2.2). Now, we take the same strategy as in [7, Theorem 3.2] and [13, Theorem 3.1]. Here, changing of variables

the integral equation (3.2) can be expressed as

Then, the solution $U$ blows up in finite time $T^{*}(U)$ such that

By a comparison argument, we obtain

Next, we shall prove a upper estimate of $T^{*}$ by two cases of $n \geq 2$ and $n = 1$ .

a) Case (a): $n \geq 2$

For $\xi \in S^{n - 1}$ and $\delta >0$ as in the theorem, we determine the sequences $\{a_j\} \subset \mathbf{R}^n$ and $\{R_j\} \subset (0,\infty)$ . Let $\{a_j\} \subset \mathbf{R}^n$ be a sequence satisfying that $|a_{j}|\to \infty$ as $j\to \infty$ , and that $a_{j} / |a_{j}| = \xi$ for any $j\in \mathbf{N}$ . Put $R_{j} = (\delta \sqrt{4 - \delta^{2}} /2)|a_{j}|$ for $\delta \in (0,\sqrt{2})$ . For $R_{j} > 0$ , let $\rho_{R_j}$ be the first eigenfunction of $-\Delta$ on

with zero Dirichlet boundary condition under the normalization

Moreover, let $\mu_{R_j}$ be the corresponding first eigenvalue. For the solutions of (1.1), we define

Then we have the following propositions.

Proposition 1. We have

and

Proof. See [25, Proposition 1].

Proposition 2. Let $0 < \alpha < 1$ and $p > 1$ . Suppose that

Then $u$ blows up in finite time, and we have

Proof. We use the method in [1, Theorem 3.7] and [3, Thorem 2.2].

By (1.1) and (3.7), we have

Since $p > 1$ and

by Jensen's inequality, we have

Thus, by (3.12)-(3.13), we obtain

By (1.3), the inequality (3.14) implies

We put $F(\zeta) = -\mu_{R_j}\zeta + \zeta^p$ . Then the function $F$ is convex in $\zeta \geq 0$ , and we get

in (3.15). $F$ is positive and increasing for all $\zeta >\mu_{R_j}^{\frac{1}{p - 1}}$ . If $w_{j}(0)$ satisfies (3.10), then (3.16) implies that $w_{j}(t) > \mu_{R_{j}}^{\frac{1}{p - 1}}$ for all $t\in (0,T^{*})$ (see [1, P.24-25]). Knowing that $w_{j}(t)\geq w_{j}(0) > \mu_{R_{j}}^{\frac{1}{p - 1}}$ for all $t\in (0,T^{*})$ , it follows from (3.16) that

Therefore the function $w_{j}(t)$ satisfying (3.17) is an upper solution of the problem

we have $w_{j}(t) \geq \zeta(t)$ by comparison principle (see [14, Theorem 2.3]).

On the other hand, since $F(0) \geq 0$ , $F(\zeta) > 0$ and $F'(\zeta) > 0$ for all $\zeta \geq w_j(0) > \mu_{R_j}^{\frac{1}{p - 1}}$ . Then, it follows from [1, Lemma 3.8](see also [23, Lemma 3.10]) that $v(t) = g\left(\frac{t^\alpha}{\Gamma(\alpha + 1)}\right)$ is a lower solution for (3.18), where $v(t)$ satisfies

and $g(t)$ solves the ordinary differential equation

By comparison principle (see [14, Theorem 2.3]), we obtain $\zeta(t) \geq v(t)$ . Solving the initial value problem (3.19), we have the solution

and obtain that $g(t) \to \infty$ as $t \to \frac{\log\left(1 - \mu_{R_j}w^{1 - p}(0)\right)}{(1 - p)\mu_{R_j}}$ . By comparison principle (see [14, Theorem 2.3]), we conclude that

By (3.20), if $w_{j}(0)$ satisfies (3.10), then we obtain that $v(t) \to \infty$ as

and that $w_{j}(t)$ blows up in finite time. Therefore, the solution $u$ blows up in finite time, and it follows that the estimate (3.11) holds, the proof of Proposition 2 is complete.

Now let us prove the Case (a).

By Propositions 1 and 2, we obtain that

From arbitrariness of $\xi \in S^{n - 1}$ and $\delta >0$ , by (3.22), we obtain

By (3.6) and (3.23), we have

Therefore, we obtain (1.10). Moreover, by (1.10) and (1.11), we have (1.12). This completes the proof.

b) Case (b): $n = 1$

Let $a_{j} = j$ or $-j$ . Put $R_{j} = j / 2$ . For $R_{j} > 0$ , let $\rho_{R_j}$ be the first eigenfunction of $-\frac{\partial^2}{\partial x^2}$ on $(-R_{j},R_{j})$ with zero Dirichlet boundary condition under the normalization

Moreover, let $\mu_{R_j}$ be the corresponding first eigenvalue. For the solutions of (1.1), we define

Then we have the following propositions.

Proposition 3. We have

and

Proof. See [25, Proposition 2].

Proposition 4. Let $0 < \alpha < 1$ and $p > 1$ . Suppose that

Then $u$ blows up in finite time, and we have

Proof. It is shown in the same way as in Proposition 2.

Finally, let us prove the Case (b). The rest of the proof is the same as in that of the Case (a).

By Propositions 3 and 4, we see that

From (3.6) and (3.30), we have

Therefore, we obtain (1.14). Moreover, by (1.14) and (1.15), we have (1.16). This completes the proof.