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Zbl 0861.35094
de Bouard, Anne
(de Bouard, Anne)
Stability and instability of some nonlinear dispersive solitary waves in higher dimension.
(English)
[J] Proc. R. Soc. Edinb., Sect. A 126, No.1, 89-112 (1996). ISSN 0308-2105; ISSN 1473-7124/e

This extensive paper is concerned with stability properties of radially symmetric solitary waves solutions for nonlinear evolution equations $$\partial_t u+\Delta\partial_{x_1}u+ \partial_{x_1}(f(u))=0,\quad x=(x_1,x')\in\bbfR^n=\bbfR\times\bbfR^{n-1},\tag1$$ and $$\partial_tu-\Delta\partial_tu+(f(u))_{x_1}=0,\quad x=(x_1,x_2)\in\bbfR^2.\tag2$$ Suppose $n$ is 2 or 3, $f\in C^{s+1}(\bbfR^n)$, $s>1+2^{-1}n$, $f(0)=f'(0)=0$, $f(s)=O(|s|^{p+1})$ as $|s|\to+\infty$, and $0<p<4(n-2)^{-1}$. The author considers a smooth function in (1) of the form $u(x,t)=\varphi_c(x_1-ct,x')$, $c>0$. Then, if $\varphi_c$ and $\Delta\varphi_c$ decrease to $0$ at infinity, we have $$-c\varphi_c+ \Delta\varphi_c+f(\varphi_c)=0.\tag3$$ Under the assumption above, the equation (3) possesses a positive, radially symmetric solution $\varphi_c\in H^1(\bbfR^n)$. The function $\varphi_c$ is called stable if for all $\varepsilon>0$, there is $\delta>0$ such that, if $u_0\in U_\delta$ and $u(.,t)$ is a solution of (1), with $u(.,0)=u_0$, then $u(.,t)\in U_\varepsilon$ for all $t>0$, where $U_\varepsilon$ is the set of $u\in H^1(\bbfR^n)$ such that $\inf_{\alpha\in\bbfR^n}|u-\varphi_c(.,-\alpha)|_1<\varepsilon$.\par The main result of this paper states that if the curve $c\mapsto\varphi_c$ is $C^1$ with values in $H^2(\bbfR^n)$, there exist $C>0$, $\delta_1>0$ such that $|{d\varphi_c\over dx} (x)|\le Ce^{-\delta_1|x|}$, $x\in\bbfR^n$, and the null space of the linearized operator $L_c=-\Delta+c-f'(\varphi_c)$ is spanned by $\{\partial_{x_j}\varphi_c; 1\le j\le n\}$, then $\varphi_c$ is stable if and only if $d''(c)>0$, where $d(c)= E(\varphi_c)+cQ(\varphi_c)$ and $E(\varphi_c)= 2^{-1}\int_{\bbfR^n}|\nabla \varphi_c|^2dx- \int_{\bbfR^n} F(\varphi_c)dx$ with $F'=f$, $F(0)=0$ and $Q(\varphi_c)= 2^{-1}\int_{\bbfR^n} \varphi^2_c dx$. Moreover, if we define the stability of $\varphi_c$ for equation (2) in the same way as we did above for the equation (1), then the same result as for (1) holds for the equation (2).
[D.M.Bors (Iaşi)]
MSC 2000:
*35Q51 Solitons
35B35 Stability of solutions of PDE
35Q30 Stokes and Navier-Stokes equations
76B15 Wave motions (fluid mechanics)

Keywords: radially symmetric solitary waves; nonlinear evolution equations

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