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Stability and instability of some nonlinear dispersive solitary waves in higher dimension. (English) Zbl 0861.35094
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 $\inf_{\alpha\in\bbfR^n}|u-\varphi_c(.,-\alpha)|_1<\varepsilon$. 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).
Reviewer: D.M.Bors (Iaşi)

35Q51Soliton-like equations
35B35Stability of solutions of PDE
35Q30Stokes and Navier-Stokes equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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