On coupled non-linear Schrödinger systems with singular source term

1.   Introduction

In this paper, we investigate the question of blow-up for nonnegative classical solution $u(x, t)$ of the following initial-boundary value problem defined in higher dimensional spaces

with the following initial-boundary conditions

In (1.1) and (1.2), $\Omega\subseteq\mathbb{R}^N, N\geq3$ is a bounded star-shaped region with smooth boundary $\partial\Omega$, $\nu$ is the outward normal vector to $\partial\Omega$. Moreover, we assume that $h\in\mathcal{C}^2(\mathbb{R}^+), 0 < h_m < h'(s) < h_M$ for $s > 0$, $u_0\in\mathcal{C}^2(\overline{\Omega})$, $f, g$ are nonnegative continuous functions. The $N\times N$ matrix $(a^{ij}(x))$ is a differential positive definite matrix; that is, there exists a constant $\theta$ such that

The question of blow-up of solutions to nonlinear parabolic equations and systems has received considerable attention in the literature. We refer to the reader the books of Straughan ^[1] and Quittner-Souplet ^[2], the survey paper of Bandle and Brunner ^[3] and the papers of Vazquez ^[4] and Weissler ^[5,6]. Most of the papers concerned with the existence and non-existence of global solutions, blow-up solutions, upper bounds on blow-up time, blow-up rates and asymptotic behavior of solutions. The blow-up phenomena of solutions to nonlinear parabolic equations and systems with nonlinear or linear Neumann boundary conditions was studied by many authors (see ^[7,8,9,10]). Some special cases of (1.1) and (1.2) have been treated already. Imai and Mochizuki ^[11] and Zhang ^[12] considered the following problem:

with different boundary conditions, where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $(N\geq2)$ with smooth boundary. Under certain assumptions on the known functions, sufficient conditions were developed for the existence of global solution or blow-up solution. Moreover, an upper bound of the blow-up time was also derived. Gao, Ding and Guo ^[13] studied the following parabolic equation

where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $(N\geq2)$ with smooth boundary. The authors obtained the conditions therefor the existence of the global solution and blow-up solution. Song and Lv ^[14,15] studied the following semilinear parabolic equation with weighted inner source terms

When the initial-boundary value problem with nonlinear Neumann boundary condition, they obtained the bounds for the blow-up time of the solution in three dimensional space (see ^[13]). In ^[14], the initial-boundary value problem with homogeneous Dirichlet or Neumann boundary condition, they derived the bounds for the blow-up rate and the blow-up time in any smooth bounded domain $\Omega\subseteq\mathbb{R}^N, N\geq3$. Recently, Ma and Fang ^[16] considered the following equation

Based on the auxiliary function method and the modified differential inequality technique, conditions on weight function and nonlinearities to guarantee the solution exists globally or blows up at finite time were established. Also, the authors derived an upper bound and a lower bound for the blow-up time. For more special cases of (1.1) and (1.2) with inner source term, one can refer to ^{[17,18,19,20]}.

In the present paper, we do not constraint $f, g$ to satisfy the the conditions in ^[16]. By constructing completely different conditions on the known data and auxiliary functions with those in the paper mentioned above, we use the modified differential inequality technique to seek the conditions which guarantee the solution of (1.1) and (1.2) exists globally or blows up in finite time. Under some certain assumption, we determine a lower bound on blow-up time in a convex bounded domain $\Omega\subset\mathbb{R}^N, N\geq3$ if blow-up occurs.

This paper is organized as follows: By establishing some appropriate auxiliary functions and using first-order differential inequalities technique, we derive a lower and an upper bounds for the blow-up time in sections 2 and 3, respectively. In section 4, we will establish the conditions to guarantee that the solution to (1.1) and (1.2) exists globally.

2.   A lower bound for the blow-up time

We list some Sobolev type inequalities which will be used in this paper.

Lemma 2.1. (see [12, p976]) Let $\Omega$ be a bounded star-shaped domain in $\mathbb{R}^N$, $N\geq2$ and $L_0 = \min_{\partial\Omega}x\cdot \textbf{n}, d = \max_{\overline{\Omega}}|x|,$ Then we have

Lemma 2.2. (see [13, Corollary IX14, p168]) Let $c_s$ be a constant depending on $\Omega$ and $N$ and $c = \sqrt{2}c_s^\frac{3}{2},$ for $N = 3$, and $c = c_s^\frac{N}{2(N-2)},$ for $N > 3$. Then we have

Moreover, we suppose that positive functions $a, f, g$ and $h$ satisfy

where $k_1, k_2, n, \beta, h_M$ are positive constants. We have the following results:

Theorem 2.1. Assume that the non-negative functions $a, f, h$ and $g$ satisfy the conditions (2.3). If the nonnegative solution $u(x, t)$ of (1.1) and (1.2) becomes unbounded in the measure $\varphi$ at some finite time $t^*$, then $t^*$ is bounded from below by

where $\sigma_1, \sigma_2, \sigma_3$ and $\sigma_4$ are positive constants and $\varphi(t)$ will be defined in (2.5).

Proof. Firstly, we define an auxiliary function

and we compute

Using (1.3) and (2.3), we have

In order to simplify our computations, we let

and rewrite (2.7) as

where we have let $h_M = 1$ for convenience. In (2.9), we have used the fact

By using Lemma 2.1, we have

Using Hölder inequality and the Young inequality for (2.11), we have

where $\varepsilon_1$ is a positive constant to be determined later. Substituting (2.12) into (2.9), we obtain

Using Hölder inequality and the Young inequality again for (2.12), we have

and

Obviously, since $n > 2(N-2)(m-1)$, $4n > 4(N-2)+(2N-3)$. Inserting (2.14) and (2.15) into (2.13), we have

where

We use the Schwarz inequality to bound

Now, we use Lemma 2.2 with $w = v^n$ for (2.16) to get

Applying the Young inequality again to obtain

for $\varepsilon_2 > 0$, inserting (2.19) back into (2.16), we obtain

Now, we choose that

to have

where

Recalling the definition of $\varphi(t)$ in (2.5), (2.23) can be rewritten as

Then

It follows on integrating (2.26) from 0 to $t$ that

so that letting $t\rightarrow t^*$, we conclude that

Thus, the proof of Theorem 2.1 is complete.

Remark 2.1. The special case $h(u) = u$ and $b(x) = 1$ in (1.1) and (1.2) was considered by ^[6], and lower bound was derived under some suitable assumption. Obviously, our result is more general.

3.   Upper bound of the blow-up time

In this section, we establish some auxiliary functions

We may have the following results:

Theorem 3.1. Let $u$ be a nonnegative solution of (1.1) and (1.2). Assume that $f, h$ and $g$ satisfy the conditions

and $H(0) > 0$. Then the solution $u(x, t)$ of problems (1.1) and (1.2) blows up in $L^2$-norm at some finite time $t^*\leq T$ with

We note that $h(s) = s^{\gamma_1}, \ k_2\geq1+\gamma_1$, $f(s) = s^{\gamma_2}, \ 2k_2(1+q)\leq1+\gamma_2$ and $g(s) = s^{\gamma_3}, \ 2k_2(1+p)\leq1+\gamma_3, \ \gamma_2, \gamma_3 > 3$, $k_3 = min\{k_2(1+p), k_2(1+q)\}$ satisfy these requirements.

Proof. From the definition of $\Phi(t)$ in (3.1), we compute

Differentiating $H(t)$ and using divergence theorem, we can derive

which implies $H(t) > 0$ for $t > 0$, since $H(0) > 0$.

Using Schwarz inequality and $(3.3)_1$, we have

Combining the above Eq (3.5), we can obtain the inequality

which may be rewritten as

Integrating (3.9) from 0 to $t$, we can have

Substituting (3.10) into (3.5) yields that

Integrating now (3.11) from 0 to $t$, we obtain the inequality

But this inequality can not hold for

In conclusion, the solution $u(x, t)$ of problems (1.1) and (1.2) fails to exist by blowing up at some finite time $t^* < T$.

Remark 3.1. If $p = q = 0$, integrating (3.11) from $t$ to $\infty$, one can see that

is valid for all $t > 0$, which implies $t^* = \infty$; that is, $T = \infty$.

In particularly, if the $N\times N$ matrix $(a^{ij}(x))$ is a diagonal matrix,

then problems (1.1) and (1.2) can be rewritten as

The last term of $(3.16)_1$ means that the system is subjected to external force. Now, we define some auxiliary functions:

where $\phi$ is the associated eigenfunction of the problem defined as

and $\lambda_1$ is the first eigenvalue of the problem (3.18). We have the following theorem:

Theorem 3.2. Let $u(x, t)$ be the nonnegative classical solution of the nonlinear parabolic problem (3.16). Assume that $f, g, h$ satisfy the conditions

and the initial condition satisfies

Then $\psi(t)$ which was defined in (3.17) must blow up at some finite time $t^*$ and the upper bound for $t^*$ can be given in the form

Proof. From (3.17) we compute

where we have used Eq (3.16), the divergence theorems (3.17), (3.18) and the assumption (3.19). Making use of Hölder's inequality and (2.3), we have

Combining (3.22) and (3.23), we obtain

Suppose that the initial data $u_0$ is large enough to insure $\Theta(\psi(0)) > 0.$ Then it can be derived from (3.24) that $\psi(t)$ is increasing for $t$ small. Since $\Theta(\psi)$ is increasing in $\psi$ from its nonnegative minimum, it follows that $\Theta(\psi(t))$ is increasing in $t$ for $t > 0$. This shows that $\psi'(t)$ remains positive, so $\psi(t)$ blows up at some time $t^*$. From (3.24), we can derive the upper bound for $t^*$:

4.   Global solution

In this section, we derive the conditions on the known functions $f$, $g$ and $h$ to guarantee existence of global solution. Our main result is the following Theorem 4.1.

Theorem 4.1. Let $u$ be nonnegative classical solution of problems (1.1) and (1.2). We first establish an auxiliary function:

Assume that

where $\gamma_1 > 0, \gamma_2 > 0, 2\gamma < k+1, \gamma > 1$. Then the solution $u$ of (1.1) and (1.2)can not blow up in finite time. that is to say there must be a global solution.

Proof. We begin with the auxiliary function $\Psi(t)$. Differentiating $\Psi(t)$, using the Eqs (1.1) and (1.2), conditions $(A1), (A2)$, (2.3) and divergence theorem, we have

By using Lemma 2.1, we obtain

where $\epsilon_1$ is a positive constant to be determined later. Substituting (4.4) into (4.3), we get

Since $2\gamma < 1+k$ and $\gamma > 1$, by Hölder's inequality, we have

and

Inserting (4.6) and (4.7) into (4.5) and choosing $\epsilon_1 = \frac{2(2p-1)h_m}{p^2\gamma_2}\theta$, we have

where

From (4.7), it follows that

In light of $2\gamma < 1+k$ and $\gamma > 1$, we can easily find that$\frac{2-2\gamma}{2p+2\gamma-2} < 0, \ \ \frac{2p+2\gamma-2}{2p} > 0$. So, combining (4.8) and (4.10), we have

From (4.11) we can conclude that $\Psi(t)$ can not blow up in finite time under the conditions (4.2). In fact, if $u(x, t)$ blows up at time $t^*$, then $\Psi(t)$ becomes unbounded when $t\rightarrow t^*$, which leads $\Psi^\frac{2-2\gamma}{2p}\rightarrow0$ and $\Psi^\frac{k+1-2\gamma}{2p}\rightarrow \infty$. So, $\Psi(t)$ decrease in some interval $[t_0, t^*)$ which follows that $\Psi(t)\leq \Psi(t_0)$. This is a contradiction.

Now, we consider a particular case of (3.16) with $g(u) = 0$

We still use the auxiliary function $\varphi(t)$ which is defined in (2.5), but with $n > 2(N-2)$. Assuming that $f, g, h$ satisfy the conditions (2.3) and $a(u)\geq\theta$ and referring to (2.8) we write for $g(u) = 0$

By Hölder and Schwarz inequalities, we have

Recalling (2.18) and (2.19), we have from (4.14)

where $a_1 = \frac{(2n+1)[n-2(N-2)]}{n(2n+2)}, a_2 = \frac{2(2n+1)(N-2)}{n(2n+2)}c, a_3 = \frac{1}{2n+2}\int_\Omega b^{2n+2}(x)dx$. Inserting (4.15) into (4.13), we have

Since $u$ has mean value zero it follows that

where $\mu$ is the first non-zero eigenvalue for

Lower bounds for $\mu$ can be found, for instance in ^[21,22]. From (4.16), we get

Now let the initial function $u_0$ and $\beta$ satisfy

It follows then that $\varphi(t)$ is initially decreasing. Since $\pi(\varphi)$ is increasing in $\varphi$, it follows that $\pi(\varphi(t))$ is decreasing in $t$ for $t > 0$. This shows that $\varphi'(t)$ remains negative. So $\varphi(t)$ can not blow up at any finite time. In fact

If we choose suitbale initial condition and $\beta$ to satisfy that

for some positive $\lambda$, then

or

This inequality shows that $\varphi(t)$ decays exponentially in time as $t\rightarrow \infty$. This is to say that $\varphi(t)$ remains bounded for $t > 0$. We have established the following theorem.

Theorem 4.2. Assume that $f, g, h$ satisfy the conditions (2.3) and $a(u)\geq\theta$. Then

(1) If the initial condition $u_0$ and $\beta$ to satisfy (4.20), then the function $\varphi(t)$ defined by (2.4) remains bounded in $L^2$.

(2) Furthermore, if the initial condition $u_0$ and $\beta$ to satisfy (4.22), $\varphi(t)$ decays exponentially to $\frac{2na_3}{\lambda}$ in time as $t\rightarrow \infty$.

5.   Conclusions

In this paper, lower bounds for the blow-up time of the solution are derived in higher dimensional spaces by virtue of the modified differential inequality. An upper bound for the blow-up time are specified under appropriate assumptions.

Acknowledgements

The work was supported national natural Science Foundation of China (Grant No. 11371175), the science foundation of GuangZhou Huashang College (Grant No. 2019HSDS28).

Conflict of interest

The authors declare that they have no competing interests.