Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity
Given the initial value problem \[ U^{\prime\prime}(t)+\| U^{\prime}(t)\|^{\alpha}U^{\prime}(t)+\nabla F(U(t))=0,\quad U(0)=U_0\in\mathbb{R}^N, \quad U^{\prime}(0)=U_1\in\mathbb{R}^N, \] where \(N\geq1\), \(\alpha>0\) and \(F:\mathbb{R}^N\rightarrow\mathbb{R}\) is a function of class \(C^2\). Introduce \(S=\{x\in\mathbb{R}^N:\nabla F(x)=0\}\) and assume that there exists \(\theta\in(0,1/2)\) such that for every \(a\in S\) there exists \(\sigma_a>0\) such that \(\|\nabla F(t)\|\geq| F(u)-F(a)|^{1-\theta}\) for all \(u\)'s with \(\| u-a\|\leq\sigma_a\). Fix \(\alpha\in[0,\theta/(1-\theta))\) and assume that \(U\in W^{2,\infty}\) is a solution of the initial value problem. Then, there exists \(C>0\) such that for all \(t>0\) one has \(\| U^{\prime}(t)\|+\| U(t)-a\|\leq C\cdot(1+t)^{-\eta}\), where \(\eta=\frac{\theta-(1-\theta)\alpha}{1-2\theta+(1-\theta)\alpha}\) (Theorem 1.3). In particular, \[ \lim_{t\rightarrow+\infty}\| U^{\prime}(t)\|+\| U(t)-a\|=0\quad\text{(Theorem 1.2)}. \] The proofs are based on several differential inequalities of first order and on the Lojasiewicz gradient inequality.
scientific article; zbMATH DE number 5314308