Abstract: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 $$ u(-\ell ,t)=u(\ell ,t)=0,\hskip 1em\relax u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,\hskip 1em\relax u(x,0)=\varphi (x), $$ or with the initial boundary value conditions $$ u_{x}(-\ell ,t)=u_{x}(\ell ,t)=0,\hskip 1em\relax u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,\hskip 1em\relax u(x,0)=\varphi (x), $$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.
Keywords: nonlinear parabolic equation, initial boundary value problem, classical global solutions
AMS Subject Classification: 35K35, 35K60