This extensive paper is concerned with stability properties of radially symmetric solitary waves solutions for nonlinear evolution equations \[ \partial_t u+\Delta\partial_{x_1}u+ \partial_{x_1}(f(u))=0,\quad x=(x_1,x')\in\mathbb{R}^n=\mathbb{R}\times\mathbb{R}^{n-1},\tag{1} \] and \[ \partial_tu-\Delta\partial_tu+(f(u))_{x_1}=0,\quad x=(x_1,x_2)\in\mathbb{R}^2.\tag{2} \] Suppose \(n\) is 2 or 3, \(f\in C^{s+1}(\mathbb{R}^n)\), \(s>1+2^{-1}n\), \(f(0)=f'(0)=0\), \(f(s)=O(|s|^{p+1})\) 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 \[ -c\varphi_c+ \Delta\varphi_c+f(\varphi_c)=0.\tag{3} \] Under the assumption above, the equation (3) possesses a positive, radially symmetric solution \(\varphi_c\in H^1(\mathbb{R}^n)\). 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 \(u\in H^1(\mathbb{R}^n)\) such that \(\inf_{\alpha\in\mathbb{R}^n}|u-\varphi_c(.,-\alpha)|_1<\varepsilon\).
The main result of this paper states that if the curve \(c\mapsto\varphi_c\) is \(C^1\) with values in \(H^2(\mathbb{R}^n)\), there exist \(C>0\), \(\delta_1>0\) such that \(|{d\varphi_c\over dx} (x)|\leq Ce^{-\delta_1|x|}\), \(x\in\mathbb{R}^n\), and the null space of the linearized operator \(L_c=-\Delta+c-f'(\varphi_c)\) is spanned by \(\{\partial_{x_j}\varphi_c; 1\leq j\leq n\}\), then \(\varphi_c\) is stable if and only if \(d''(c)>0\), where \(d(c)= E(\varphi_c)+cQ(\varphi_c)\) and \(E(\varphi_c)= 2^{-1}\int_{\mathbb{R}^n}|\nabla \varphi_c|^2dx- \int_{\mathbb{R}^n} F(\varphi_c)dx\) with \(F'=f\), \(F(0)=0\) and \(Q(\varphi_c)= 2^{-1}\int_{\mathbb{R}^n} \varphi^2_c dx\). 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).