A reaction–diffusion approximation of a semilinear wave equation with damping

Abstract

A reaction–diffusion approximation is a method that solutions of multi-component reaction–diffusion systems approximate those of differential equations. We introduce the reaction–diffusion approximations of a semilinear wave equation and a semilinear damped wave equation under some assumptions of a reaction term. These approximation systems consist of a two-component reaction–diffusion system with a small parameter. In this paper, we prove that a first component of a solution for the system converges to a solution for the semilinear damped wave equation as the parameter tends to zero. Moreover, let us show the numerical results of reaction–diffusion approximation for the wave equation and the damped wave equation, respectively.

Appendix: Proof of Proposition 1

Proof

If $(v_1,v_2)$ is a smooth solution of (13) with respect to time $t \in [0,T]$ and spatial variable $x \in {\mathbb {R}}^N$, then $v_1$ satisfies the Eq. (14). By a priori estimate of (14), we shall prove Proposition 1. Multiplying (14) by $v_{1,t}$ and integrating it over ${\mathbb {R}}^N$, we calculate it as follows:

$$\begin{aligned}&\frac{d}{dt} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx +\frac{d_1d_2}{2} \varepsilon ^2 \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx +\frac{d + ad_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx -\int _{{\mathbb {R}}^N} F(v_1) \,dx \right] \\&\quad = -a \int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx -( d_1 +d_2 ) \varepsilon \int _{{\mathbb {R}}^N} | \nabla v_{1,t} |^2 \,dx \, \le \, 0. \end{aligned}$$

Here, $a \ge 0$ was used. Integrating the above differential inequality over [0, t], we see that

$$\begin{aligned}&\frac{1}{2} \int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx +\frac{d_1d_2}{2} \varepsilon ^2 \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx +\frac{d + ad_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx -\int _{{\mathbb {R}}^N} F(v_1) \,dx \le C, \end{aligned}$$

where C is a positive constant depending on the initial data $w_0 \in H^2({\mathbb {R}}^N)$ and $w_1 \in L^2({\mathbb {R}}^N)$. Since $-f_3 |v_1|^2 \le -F(v_1)$ from (7), it holds that

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx +\frac{d_1d_2}{2} \varepsilon ^2 \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx +\frac{d + ad_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx \le f_3 \int _{{\mathbb {R}}^N} v_1^2 \,dx +C. \end{aligned} \end{aligned}$$

(22)

We note that $v_{1,t}$, $\varepsilon \varDelta v_1$ and $|\nabla v_1|$ are bounded in $L^2({\mathbb {R}}^N)$ and $v_1$ is bounded in $L^{p+1}({\mathbb {R}}^N)$ for any $t \in [0,T]$ and any $\varepsilon \in (0,\varepsilon _0)$ if the $L^2$-norm of $v_1$ is bounded for any $t \in [0,T]$ and any $\varepsilon \in (0,\varepsilon _0)$.

Next, multiplying (14) by $v_1$ and integrating it over [0, t], we see that

$$\begin{aligned}&\frac{d^2}{dt^2} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} v_1^2 \,dx \right] +\frac{d}{dt} \left[ \frac{(d_1+d_2) \varepsilon }{2} \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx \right] \\&\quad = -a \int _{{\mathbb {R}}^N} v_1 v_{1,t} \,dx +\int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx -d_1d_2\varepsilon ^2 \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx -(d+ad_1\varepsilon ) \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx +\int _{{\mathbb {R}}^N} f(v_1)v_1 \,dx. \end{aligned}$$

Since (6) and $|v_1v_{1,t}| \le \frac{1}{2} ( v_1^2 +v_{1,t}^2 )$ hold, we estimate the above identity as follows:

$$\begin{aligned}&\frac{d^2}{dt^2} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} v_1^2 \,dx \right] +\frac{d}{dt} \left[ \frac{(d_1+d_2) \varepsilon }{2} \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx \right] \\&\quad \le C\int _{{\mathbb {R}}^N} v_1^2 \,dx +C\int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx -d_1d_2\varepsilon ^2 \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx -(d+ad_1\varepsilon ) \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx \\&\quad \le C\int _{{\mathbb {R}}^N} v_1^2 \,dx +C\int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx. \end{aligned}$$

From (22), We note $\int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx \le C\int _{{\mathbb {R}}^N} v_1^2 \,dx+C$. Hence, we have that

$$\begin{aligned}&\frac{d^2}{dt^2} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} v_1^2 \,dx \right] +\frac{d}{dt} \left[ \frac{(d_1+d_2) \varepsilon }{2} \int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx \right] \le C \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} v_1^2 \,dx \right] +C, \end{aligned}$$

where C is depending on $f_1,f_3,a,T,\varepsilon _0$ and the initial data $w_0,w_1$. By [16, Lemma 2.2], there exists a constant $C > 0$ such that for any $t \in [0,T]$ and any $\varepsilon \in (0,\varepsilon _0)$,

$$\begin{aligned} \int _{{\mathbb {R}}^N} v_1(t,x)^2 \,dx \le C. \end{aligned}$$

(23)

Therefore, by (22) and (23), we obtained the following boundedness with respect to $v_1$ for any $t \in [0,T]$ and any $\varepsilon \in (0,\varepsilon _0)$:

$$\begin{aligned}&\int _{{\mathbb {R}}^N} v_1^2 \,dx,&\int _{{\mathbb {R}}^N} | \nabla v_1 |^2 \,dx,&\int _{{\mathbb {R}}^N} v_{1,t}^2 \,dx \quad \text { and}&\varepsilon ^2 \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx. \end{aligned}$$

(24)

From here, we shall show the boundedness of $\int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx$ and $\int _{{\mathbb {R}}^N} (\varDelta v_1)^2 \,dx$ for any t and $\varepsilon$. Multiplying (14) by $-\varDelta v_{1,t}$ and integrating it over ${\mathbb {R}}^N$ yield that

$$\begin{aligned}&\frac{d}{dt} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{d_1d_2 \varepsilon ^2}{2} \int _{{\mathbb {R}}^N} |\nabla ( \varDelta v_1)|^2 \,dx +\frac{d+d_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} ( \varDelta v_1)^2 \,dx \right] \\& \quad = -a \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx -(d_1 +d_2) \varepsilon \int _{{\mathbb {R}}^N} ( \varDelta v_{1,t} )^2 \,dx +\int _{{\mathbb {R}}^N} f^{\prime }(v_1) \nabla v_1 \cdot \nabla v_{1,t} \,dx. \end{aligned}$$

Since $a \ge 0$, $|f^{\prime }(v_1) \nabla v_1 \cdot \nabla v_{1,t}| \le \frac{1}{2} ( |\nabla v_{1,t}|^2 +| f^{\prime }(v_1)|^2 |\nabla v_1|^2 )$ and (8) hold, we estimate the previous equation as follows:

$$\begin{aligned}&\frac{d}{dt} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{d_1d_2 \varepsilon ^2}{2} \int _{{\mathbb {R}}^N} |\nabla ( \varDelta v_1)|^2 \,dx +\frac{d+d_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} ( \varDelta v_1)^2 \,dx \right] \\&\quad \le \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{1}{2} \int _{{\mathbb {R}}^N} | f^{\prime }(v_1)|^2 |\nabla v_1|^2 \,dx \\&\quad \le \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{1}{2} \int _{{\mathbb {R}}^N} \left( f_4+f_5 |v_1|^{p-1} \right) ^2 |\nabla v_1|^2 \,dx \\&\quad \le \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +f_4^2 \int _{{\mathbb {R}}^N} |\nabla v_1|^2 \,dx +f_5^2 \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx. \end{aligned}$$

Since the integrals in (24) are bounded for t and $\varepsilon$, we have that

$$\begin{aligned} \begin{aligned}&\frac{d}{dt} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{d_1d_2 \varepsilon ^2}{2} \int _{{\mathbb {R}}^N} |\nabla ( \varDelta v_1)|^2 \,dx +\frac{d+d_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} ( \varDelta v_1)^2 \,dx \right] \\&\quad \le \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +f_5^2 \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx +C. \end{aligned} \end{aligned}$$

(25)

Here, let us show

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx \le C +C \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx. \end{aligned}$$

(26)

(i) The case of $N \ge 3$ and $2 \le N(p-1) \le \frac{2N}{N-2}$: Note that this case satisfies the condition (10). By the Hölder inequality and the Sobolev inequality, we can estimate the integral $\int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx$ as follows:

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx&\le \left( \int _{{\mathbb {R}}^N} |v_1|^{N(p-1)} \,dx \right) ^{\frac{2}{N}} \left( \int _{{\mathbb {R}}^N} |\nabla v_1|^{\frac{2N}{N-2}} \,dx \right) ^{\frac{N-2}{N}} \\&\le \Vert v_1 \Vert _{L^{N(p-1)}({\mathbb {R}}^N)}^{2(p-1)} \, \cdot \, C_S \Vert \nabla v_1 \Vert _{H^1({\mathbb {R}}^N)}^2. \end{aligned}$$

Here, by (24) and the Sobolev inequality, we have the boundedness of $\Vert v_1\Vert _{H^1({\mathbb {R}}^N)}$ for $t,\varepsilon$, and hence $\Vert v_1\Vert _{L^{\frac{2N}{N-2}}({\mathbb {R}}^N)}$ are bounded. From the boundedness of $\Vert v_1\Vert _{L^2({\mathbb {R}}^N)}$ and $\Vert v_1\Vert _{L^{ \frac{2N}{N-2} }({\mathbb {R}}^N)}$ and

$$\begin{aligned} 2 \le N(p-1) \le \frac{2N}{N-2}, \end{aligned}$$

we see that $\Vert v_1 \Vert _{L^{N(p-1)}({\mathbb {R}}^N)}^{2(p-1)}$ is bounded. Moreover, by a calculation of [, p. 302], it follows that16

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx&\le C \Vert \nabla v_1 \Vert _{H^1({\mathbb {R}}^N)}^2 = C \left( \Vert \nabla v_1 \Vert _{L^2({\mathbb {R}}^N)}^2 +\Vert D^2 v_1 \Vert _{L^2({\mathbb {R}}^N)}^2 \right) \\&\le C \left( \Vert \nabla v_1 \Vert _{L^2({\mathbb {R}}^N)}^2 +\Vert \varDelta v_1 \Vert _{L^2({\mathbb {R}}^N)}^2 \right) \end{aligned}$$

where $\Vert D^2 v_1 \Vert _{L^2({\mathbb {R}}^N)}^2 = \int _{{\mathbb {R}}^N} |D^2 v_1|^2 \,dx$ and $|D^2 v_1|^2 = \sum _{i,j=1}^{N} |\frac{\partial ^2 v_1}{\partial x_i \partial x_j}|^2$. Hence, since $\Vert \nabla v_1 \Vert _{L^2({\mathbb {R}}^N)}^2$ is bounded, it is obtained that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx&\le C+C \Vert \varDelta v_1 \Vert _{L^2({\mathbb {R}}^N)}^2. \end{aligned}$$

We complete the proof of (26) in the case of $N \ge 3$ and $2 \le N(p-1) \le \frac{2N}{N-2}$.

(ii) The case of $N \ge 3$ and $0< N(p-1) < 2$: Note that $1-\frac{2}{N}< 2-p < 1$ and $2< \frac{2}{2-p} < \frac{2N}{N-2}$. By the Hölder inequality and the Sobolev inequality, we have that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx&\le \left( \int _{{\mathbb {R}}^N} |v_1|^2 \,dx \right) ^{p-1} \left( \int _{{\mathbb {R}}^N} |\nabla v_1|^{\frac{2}{2-p}} \,dx \right) ^{2-p} \\&\le \Vert v_1\Vert _{L^2({\mathbb {R}}^N)}^{2(p-1)} \, \Vert \nabla v_1 \Vert _{L^{\frac{2}{2-p}}({\mathbb {R}}^N)}^2 \\&\le \Vert v_1\Vert _{L^2({\mathbb {R}}^N)}^{2(p-1)} \, \cdot \, C_S \Vert \nabla v_1 \Vert _{H^1({\mathbb {R}}^N)}^2 \end{aligned}$$

By a similar calculation to the latter half of (i) and the boundedness of $\Vert v_1\Vert _{L^2({\mathbb {R}}^N)}$ and $\Vert \nabla v_1\Vert _{L^2({\mathbb {R}}^N)}$, we have $\int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx \le C+C\Vert \varDelta v_1 \Vert _{L^2({\mathbb {R}}^N)}^2$ for $N \ge 3$ and $0< N(p-1) < 2$. Hence, (26) is shown in the case of $N \ge 3$ and $0< N(p-1) < 2$.

(iii) The case of $N=1,2$: By using the Hölder inequality and the Sobolev inequality: $\Vert u \Vert _{L^r({\mathbb {R}}^N)} \le C_S \Vert u \Vert _{H^1({\mathbb {R}}^N)}$ for $2 \le r < \infty$, we have that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx&\le \left( \int _{{\mathbb {R}}^N} |v_1|^{2(p+1)} \,dx \right) ^{\frac{p-1}{p+1}} \left( \int _{{\mathbb {R}}^N} |\nabla v_1|^{p+1} \,dx \right) ^{\frac{2}{p+1}} \\&\le \Vert v_1\Vert _{L^{2(p+1)}({\mathbb {R}}^N)}^{2(p-1)} \, \Vert \nabla v_1 \Vert _{L^{p+1}({\mathbb {R}}^N)}^2 \\&\le \Vert v_1\Vert _{L^{2(p+1)}({\mathbb {R}}^N)}^{2(p-1)} \, \cdot \, C_S \Vert \nabla v_1 \Vert _{H^1({\mathbb {R}}^N)}^2 \\&\le C \Vert v_1\Vert _{H^1({\mathbb {R}}^N)}^{2(p-1)} \, \cdot \, \Vert \nabla v_1 \Vert _{H^1({\mathbb {R}}^N)}^2 \end{aligned}$$

From (24), $\Vert v_1\Vert _{H^1({\mathbb {R}}^N)}$ is bounded for any t and any $\varepsilon$, so that it holds that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx \le C \Vert \nabla v_1 \Vert _{H^1({\mathbb {R}}^N)}^2 \end{aligned}$$

By a similar discussion to the above, we proved that for $N=1,2$,

$$\begin{aligned} \int _{{\mathbb {R}}^N} |v_1|^{2(p-1)} \, |\nabla v_1|^2 \,dx \le C +C \int _{{\mathbb {R}}^N} ( \varDelta v_1 )^2 \,dx. \end{aligned}$$

Thus, (26) was shown.

Next, by (25) and (26), it follows that

$$\begin{aligned}&\frac{d}{dt} \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{d_1d_2 \varepsilon ^2}{2} \int _{{\mathbb {R}}^N} |\nabla ( \varDelta v_1)|^2 \,dx +\frac{d+d_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} ( \varDelta v_1)^2 \,dx \right] \\&\quad \le C \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{d+d_1 \varepsilon }{2}\int _{{\mathbb {R}}^N} |\varDelta v_1|^2 \,dx \right] +C \\&\quad \le C \left[ \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx +\frac{d_1d_2 \varepsilon ^2}{2} \int _{{\mathbb {R}}^N} |\nabla ( \varDelta v_1)|^2 \,dx +\frac{d+d_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} ( \varDelta v_1)^2 \,dx \right] +C. \end{aligned}$$

Putting

$$\begin{aligned} X(t) = \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}(t,x)|^2 \,dx +\frac{d_1d_2 \varepsilon ^2}{2} \int _{{\mathbb {R}}^N} |\nabla ( \varDelta v_1(t,x))|^2 \,dx +\frac{d+d_1 \varepsilon }{2} \int _{{\mathbb {R}}^N} ( \varDelta v_1(t,x))^2 \,dx, \end{aligned}$$

we see that $\frac{dX}{dt} \le CX +C$ for time t. Solving this differential inequality with respect to t, we have that $X(t) \le X(0) e^{Ct} +e^{Ct}-1$ for $t \in [0,T]$. Since X(0) is a positive constant depending on the initial data $w_0 \in H^3({\mathbb {R}}^N)$ and $w_1 \in H^1({\mathbb {R}}^N)$ and t belongs to the interval [0, T], we obtained that X(t) is bounded for any $t \in [0,T]$ and any $\varepsilon \in (0,\varepsilon _0)$. Hence, it is proved that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla v_{1,t}|^2 \,dx \qquad \text { and } \qquad \int _{{\mathbb {R}}^N} (\varDelta v_1)^2 \,dx \end{aligned}$$

are bounded for any $t \in [0,T]$ and any $\varepsilon \in (0,\varepsilon _0)$. Consequently, we complete the proof of Proposition 1. $\square$