##
**Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms.**
*(English)*
Zbl 1232.35008

The authors study the initial-boundary-value problem of the following form:
\[
\begin{cases} u_{tt} - \text{div} (g(|\nabla u |^2) \nabla u) + |u_t|^{m-1} u_t=f_1(u,v), \qquad (x,t) \in \Omega \times (0,T),\\ v_{tt} - \text{div} (g(|\nabla v |^2) \nabla v) + |v_t|^{r-1} v_t=f_2(u,v), \qquad (x,t) \in \Omega \times (0,T),\\ u(x,t) = v(x,t)=0, \qquad \qquad \qquad \qquad \qquad \qquad x \in \partial \Omega \times (0,T)\\ u(x,0) = u_0(x), u_t(x,0) = u_1(x), \qquad \qquad \qquad \qquad x \in \Omega\\ v(x,0) = v_0(x), v_t(x,0) = v_1(x), \qquad \qquad \qquad \qquad x \in \Omega \end{cases}\tag{1}
\]
where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with a smooth boundary \(\partial \Omega\), \(m,r \geq 1\), and \(f_i(.,.):\mathbb R^2 \rightarrow\mathbb R\;(i=1,2)\),
\[
f_1 (u,v) = [a|u+v|^{2(p+1)} (u+v)+b|u|^p u|v|^{(p+2)}],
\]

\[ f_2 (u,v) = [a|u+v|^{2(p+1)} (u+v)+b|u|^{(p+2)} |v|^p v], \]

\[ \begin{cases} a,b > 0 \text{ are constants and } p \text{ satisfies } p > -1\;(\text{if } n=1,2), \\ -1 <p \leq \frac{4-n}{n-2},\;(\text{if } n \geq3), g \in C^1, g(s) > 0, g(s) - 2q'(s) >0, s>0. \end{cases}\tag{2} \] The authors improve the global non-existence result of problem (1) (proposed by J. Wu, S. Li and S. Chai [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 11, 3969–3975 (2010; Zbl 1190.35145)]) for a large class of initial data for which the initial energy can take positive values.

The main result is: If \(2(p+2) > \max \{2q+2, m+1, r+1\}\) and the assumption (2) holds, then any solution of (1) with initial data satisfying \((\int_\Omega (G(| \nabla u_0|^2)+G (|\nabla v_0|^2))dx)^{1/2} >\alpha_1, E (0) >E_2\) cannot exist for all time, where the constants \((\alpha_1, E_1)\) are defined by the expressions \(\alpha_1 = B^{-(p+2/(p+1))}\), \(B=\eta^{1/(2(p+2))}\). A precise proof of a global nonexistence theorem for certain solutions with positive initial energy is presented.

\[ f_2 (u,v) = [a|u+v|^{2(p+1)} (u+v)+b|u|^{(p+2)} |v|^p v], \]

\[ \begin{cases} a,b > 0 \text{ are constants and } p \text{ satisfies } p > -1\;(\text{if } n=1,2), \\ -1 <p \leq \frac{4-n}{n-2},\;(\text{if } n \geq3), g \in C^1, g(s) > 0, g(s) - 2q'(s) >0, s>0. \end{cases}\tag{2} \] The authors improve the global non-existence result of problem (1) (proposed by J. Wu, S. Li and S. Chai [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 11, 3969–3975 (2010; Zbl 1190.35145)]) for a large class of initial data for which the initial energy can take positive values.

The main result is: If \(2(p+2) > \max \{2q+2, m+1, r+1\}\) and the assumption (2) holds, then any solution of (1) with initial data satisfying \((\int_\Omega (G(| \nabla u_0|^2)+G (|\nabla v_0|^2))dx)^{1/2} >\alpha_1, E (0) >E_2\) cannot exist for all time, where the constants \((\alpha_1, E_1)\) are defined by the expressions \(\alpha_1 = B^{-(p+2/(p+1))}\), \(B=\eta^{1/(2(p+2))}\). A precise proof of a global nonexistence theorem for certain solutions with positive initial energy is presented.

Reviewer: Jan Lovíšek (Bratislava)

### MSC:

35A01 | Existence problems for PDEs: global existence, local existence, non-existence |

35B44 | Blow-up in context of PDEs |

35G50 | Systems of nonlinear higher-order PDEs |

35L05 | Wave equation |

### Keywords:

global nonexistence; positive initial-energy; coupled nonlinear wave equation; damping; source terms; Holder’s inequality; Young’s inequality### Citations:

Zbl 1190.35145
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\textit{F. Liang} and \textit{H. Gao}, Abstr. Appl. Anal. 2011, Article ID 760209, 14 p. (2011; Zbl 1232.35008)

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