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Global existence of solutions to a fully nonlinear fourth-order parabolic equation in exterior domains. (English) Zbl 0762.35042
The purpose of this paper is the investigation of the existence of smooth solutions $$u=u(t,x)$$ to the following nonlinear boundary value problem $u_ t+\Delta^ 2u=f(\overline\nabla^ 4u)\quad\text{in }R^ +_ 0\times\Omega,\quad u(t=0)=u_ 0\quad\text{in }\Omega,\quad u=\Delta u=0\quad\text{on }R^ +_ 0\times\partial\Omega,$ where $$\overline\nabla^ 4u=(\nabla^ \alpha u)_{0\leq|\alpha|\leq 4}$$ and $$\Omega$$ is an exterior domain in $$R_ n$$. Uniqueness results are also given as well as decay rates on the solution as $$t\to\infty$$.

##### MSC:
 35K35 Initial-boundary value problems for higher-order parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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