Global existence and blow up of the solution for nonlinear Klein-Gordon equation with general power-type nonlinearities at three initial energy levels
The authors consider the Cauchy problem for a nonlinear Klein-Gordon system: \[ \begin{aligned} & u_{tt}-\Delta u + u=f(u),\qquad x\in\mathbb{R}^n,\qquad t\in(0,+\infty), \\ &u(x,0)=u_0(x),\qquad u_t(x,0)=u_1(x),\qquad x\in\mathbb{R}^n, \end{aligned}\tag{1} \] where \(u_0\in H^1(\mathbb{R}^n)\), \(u_1\in L_2(\mathbb{R}^n)\). The main results are sufficient conditions on the initial data such that the weak solution exists globally or blows up in a finite time. The main assumption is as follows: \[ \begin{aligned} & f\in C^1,\qquad u(uf'(u)-pf(u))\geqslant 0\quad\text{for all}\quad u,\quad p>1, \\ & |f(u)|\leqslant \sum_{k=1}^{l} a_k|u|^{p_k},\quad a_k>0,\quad 1\leqslant k<l,\quad \\ & 1<p_l<p_{l-1}<\ldots<p_1\leqslant p<\frac{n}{n-2}\quad \text{if}\quad n\geqslant 3, \\ & 1<p_l<p_{l-1}<\ldots<p_1\leqslant p<\infty\quad \text{if}\quad n=3. \end{aligned} \] The main results read as follows. Theorem. There exist \(T>0\) and a unique solution to problem (1) over \(t\in [0,T]\). Moreover, if \(T_{max} = \sup\{T > 0 : u = u(t) \text{ exists on } [0, T]\} <\infty\), then \(\lim_{t\to T_{max}} \|u(t)\|_{L_q(\mathbb{R}^n)}=\infty\) for all \(n\geqslant 1\) such that \(q >(p-1)n/2\). Denote \[ \begin{aligned} &J(u):=\frac{1}{2}\|u\|_{H^1(\mathbb{R}^n)}^2-\int_{\mathbb{R}^n} F(u) dx,\quad I(u):=\frac{1}{2}\|u\|_{H^1(\mathbb{R}^n)}^2-\int_{\mathbb{R}^n} u f(u) dx, \\ & E(t):=\frac{1}{2}\|u_t\|_{L_2(\mathbb{R}^n)} + \frac{1}{2}\|u\|_{L_2(\mathbb{R}^n)} -\int_{\mathbb{R}^n} F(u) dx,\quad F(u):=\int_{0}^{u}f(s) d s, \\ &d=\inf_{u\in\mathcal{N}} J(u),\quad \mathcal{N}:=\{u\in H^1(\mathbb{R}^n): I(u)=0,\ u\ne0\}, \\ &W:=\{u\in H^1(\mathbb{R}^n): I(u)>0,\ J(u)<d\}\cup\{0\}. \end{aligned} \] Theorem. Suppose that \(0 < E(0) < d\) and \(I(u_0) > 0\) or \(\|u_0\|_{H^1(\mathbb{R}^n)} = 0\). Then problem (1) admits a global weak solution \(u(t) \in C((0,\infty); H^1(\mathbb{R}^n))\), \(u_t(t)\in C((0,\infty); L_2(\mathbb{R}^n))\) and \(u(t)\in W\) for \(0\leqslant t <\infty\). Theorem. Suppose that \(E(0) = d\) and \(u_0\in W\), then problem (1) admits a global weak solution \(u(t)\in C([0, T ); H^1(\mathbb{R}^n))\) with \(u_t(t)\in C([0, T); L_2(\mathbb{R}^n))\). Theorem. Assume that \(E(0) > 0\), \(I(u_0) < 0\), \((u_0, u_1)_{L_2(\mathbb{R}^n)} \geqslant 0\) and \[ \|u_0\|_{L_2(\mathbb{R}^n})^2>\frac{2(p+1)}{p-1}E(0)>0. \] Then the corresponding solution \(u\) of problem (1) blows up in a finite time.
scientific article; zbMATH DE number 7074564
scientific article