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On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation. (English) Zbl 0893.35078

The following integro-differential equations are studied \[ u_{tt}-\bigl(a+b \| \nabla u\|^{2\gamma}_{L^2}\bigr) \Delta u+| u_t|^\beta u_t=f(u), \] \(u=u(t,x)\), \(x\in \Omega\) bounded in \(\mathbb{R}^n\), \(f(u)\) typically like \(\pm| u|^\alpha u\), together with initial conditions \(\partial^j_t u(t=0)=u_j\), \(j=1,2,\) and Dirichlet boundary conditions on \(\partial \Omega\), \(a,b\geq 0\), \(a+b>0\), \(\gamma\geq 1\), \(\alpha>0\). For \(a>0\) a global existence theorem is proved for certain \(\alpha,\beta, \gamma\) if the initial first (not necessarily the higher) energy is sufficiently small. For \(f(u)=| u|^\alpha u\), \(a>\max \{\beta, 2\gamma \}\) a blow-up theorem is presented if the initial first energy is negative, and in some cases the life-span is estimated. For the global existence proof, a so-called modified potential method is used.
Reviewer: R.Racke (Konstanz)

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
45K05 Integro-partial differential equations
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