Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation. (English) Zbl 1233.35145
Summary: We consider the nonlinear Kirchhoff-type equation
\[
u_{tt} + M\left(\left\| {D^m u(t)} \right\|_2^2\right) (-\Delta)^m u + |{u_t}|^{q-2} u_t = |{u_t}|^{p-2} u
\]
with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial data, we prove that the solution blows up in finite time.
MSC:
| 35L77 | Higher-order quasilinear hyperbolic equations |
| 35L35 | Initial-boundary value problems for higher-order hyperbolic equations |
| 35B44 | Blow-up in context of PDEs |
| 35R09 | Integro-partial differential equations |
Keywords:
nonlinear Kirchhoff-type equation; positive upper bounded initial energy; homogeneous boundary conditionsReferences:
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