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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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[1] Adams, R., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[2] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Communs pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401
[3] Klainerman, S., Long-time behavior of solutions to nonlinear evolution equations, Archs ration. mech. analysis, 78, 73-98, (1982) · Zbl 0502.35015
[4] Ladyzhenskaya, O.A., The boundary value problems of mathematical physics, () · Zbl 0164.12501
[5] Leis, R., Initial boundary value problems in mathematical physics, () · Zbl 0414.73082
[6] Matsumura, A., An energy method for the equations of motion of compressible viscous and heat-conductive fluids, (), MRC
[7] Magnus, W.; Oberhettinger, F.; Soni, R.P., Formulas and theorems for the special functions of mathematical physics, (1966), Springer Berlin · Zbl 0143.08502
[8] Morawetz, C.S.; Ludwig, D., An inequality for the reduced wave operator and the justification of geometrical optics, Communs pure appl. math., 21, 187-203, (1968) · Zbl 0157.18701
[9] Murray, J.D., On a mechanical model for morphogenesis: mesenchymal patterns, Lecture notes in biomathematics, 55, 279-291, (1984)
[10] Ponce, G., Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear analysis, 9, 399-418, (1985) · Zbl 0576.35023
[11] Racke, R., L^{p}−lq-estimates for solutions to the equations of linear thermoelasticity in exterior domains, Asymptotic analysis, 3, 105-132, (1990) · Zbl 0724.35019
[12] Racke, R., Decay rates for solutions of damped systems and generalized Fourier transforms, J. reine angew. math., 412, 1-19, (1990) · Zbl 0718.35015
[13] Ramm, A.G., Scattering by obstacles, (1986), Reidel Dordrecht · Zbl 0607.35006
[14] Shibata, Y., On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domai n, Tsukuba J. math., 7, 1, 1-68, (1983) · Zbl 0524.35071
[15] Shibata, Y.; Tsutsumi, Y., Global existence theorem for nonlinear wave equation in exterior domain, Lecture notes in numerical applied analysis, 6, 155-196, (1983)
[16] Shibata, Y.; Tsutsumi, Y., On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z., 191, 165-199, (1986) · Zbl 0592.35028
[17] Truesdell, C.; Noll, W., ()
[18] Vainberg, B.R., On the short wave asymptotic behavior of solutions of stationary problems and the asymptotic behavior as t→∞ of solutions of non-stationary pr oblems, Russian math. surv., 30, 2, 1-58, (1975) · Zbl 0318.35006
[19] Werner, P., Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces, Math. meth. appl. sci., 8, 134-156, (1986) · Zbl 0609.35028
[20] Wilcox, C.H., Scattering theory for the d’Alembert equation in exterior domains, () · Zbl 0125.46005
[21] Zheng, S., Remarks on global existence for nonlinear parabolic equations, Nonlinear analysis, 10, 107-114, (1986) · Zbl 0595.35059
[22] Zheng, S., Global solutions to the second initial boundary value problem for fully nonlinear parabolic equations, Acta math. sinica, 3, 3, 237-246, (1987) · Zbl 0651.35042
[23] Zheng, S., Global existence of solutions to initial boundary value problem for nonlinear parabolic equations, Acta math. sci., 7, 4, 361-374, (1987) · Zbl 0665.35041
[24] Zheng, S.; Chen, Y., Global existence for nonlinear parabolic equations, Chin. ann. math., 7B, 1, 57-73, (1986) · Zbl 0603.35049
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