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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
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