Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
nonlinear parabolic equation; initial boundary value problem; classical global solutions
nonlinear parabolic equation; initial boundary value problem; classical global solutions
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 $$ u(-\ell ,t)=u(\ell ,t)=0,\quad u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,\quad u(x,0)=\varphi (x), $$ or with the initial boundary value conditions $$ u_{x}(-\ell ,t)=u_{x}(\ell ,t)=0,\quad u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,\quad u(x,0)=\varphi (x), $$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.
References:
[1] Cohen D.S., Murray J.D.: A generalized diffusion model for growth and dispersal in population. J. Math. Biol. 12 (1981), 237-249. MR 0624561
[2] Naimark M.A.: Linear Differential Operators. Moscow, 1954. MR 0067292 | Zbl 0227.34020
[3] Maz'ja V.G.: Sobolev Spaces. Springer-Verlag, 1985. MR 0817985 | Zbl 0692.46023
[4] Zhou Yulin, Fu Hongyuan: The nonlinear hyperbolic systems of higher order of generalized Sine-Gordon type (in Chinese). Acta Math. Sinica 26 (1983), 234-249. MR 0694886
[5] Chen Guowang: First boundary problems for nonlinear parabolic and hyperbolic coupled systems of higher order. Chinese Journal of Contemporary Mathematics 9 (1988), 98-116. MR 0997346
[6] Liu Baoping, Pao C.V.: Integral representation of generalized diffusion model in population problems. Journal of Integral Equations 6 (1984), 175-185. MR 0733043
[7] Chen Guowang: Initial value problem for a class of nonlinear parabolic system of fourth- order. Acta Mathematica Scientia 11 (1991), 393-400. MR 1174369
Search
Partner of
