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Classical global solutions of the initial boundary value problems for a class of nonlinear parabolic equations. (English) Zbl 0808.35049
Summary: The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $u_ t=- A(t)u_{x^ 4}+ B(t) u_{x^ 2}+ g(u)_{x^ 2}+ f(u)_ x+ h(u_ x)_ x+ G(u)$ with the initial boundary value conditions $$u(-\ell, t)=u(\ell, t)=0$$, $$u_{x^ 2} (-\ell,t)= u_{x^ 2}(\ell, t)=0$$, $$u(x,0)= \varphi(x)$$, or with the initial boundary value conditions $$u_ x (-\ell,t)= u_ x(\ell,t) =0$$, $$u_{x^ 3} (-\ell,t)= u_{x^ 3} (\ell,t)=0$$, $$u(x,0)= \varphi(x)$$, are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

##### MSC:
 35K35 Initial-boundary value problems for higher-order parabolic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs
##### Keywords:
classical global solutions
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