×

zbMATH — the first resource for mathematics

Parabolic problems with nonlinear boundary conditions and critical nonlinearities. (English) Zbl 0938.35077
The authors study the nonlinear parabolic problem \[ \begin{gathered} \frac {\partial u}{\partial t} =\Delta u + f(u) \text{ in } \Omega\times(0,T), \\ \frac {\partial u}{\partial \nu} = g(u) \text{ on } \partial\Omega\times(0,T), \\ u(\cdot,0) = u_0 \text{ in } \Omega, \end{gathered} \] where \(\Omega \subset \mathbb R^N\) has unit outer normal \(\nu\), and \(u\) depends on \(x\) and \(t\). Their main theorem is that if there are constants \(q \in (1,\infty)\) and \(C\) such that \[ \begin{aligned} |f(u)-f(v)|&\leq C|u-v|(|u|^{2q/N}+|v|^{2q/N} +1),\\ |g(u)-g(v)|&\leq C|u-v|(|u|^{q/N}+|v|^{q/N} +1) \end{aligned} \tag{*} \] (if \(N \geq 2\)), and if \(u_0 \in L^q(\Omega)\), then this problem has a unique solution. (If \(N=1\), the exponents in the inequality for \(g\) must be less than \(q\).) They also show that if \(u_0 \in W^{1,q}\) with \(q \in(1,\infty)\), then the condition (*) can be relaxed. These results include all known results on existence for problems with critical growth. The method of proof relies on various interpolation theorems for scales of Banach spaces.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Adams, R., Sobolev spaces, (1978), Academic Press Boston
[2] Alikakos, N.D., Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in Lp, J. differential equations, 39, 311-344, (1981) · Zbl 0501.35045
[3] Amann, H., On abstract parabolic fundamental solutions, J. math. soc. Japan, 39, 93-116, (1987) · Zbl 0616.47032
[4] Amann, H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Schmeisser/Triebel: function spaces, differential operators and nonlinear analysis, Teubner texte zur Mathematik, 133, (1993), Teubner Stuttgart, p. 9-126 · Zbl 0810.35037
[5] Amann, H., Linear and quasilinear parabolic problems. abstract linear theory, (1995), Birkhäuser Basel · Zbl 0819.35001
[6] J. Arrieta, and, A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, to appear in, Trans. Am. Math. Soc. · Zbl 0940.35119
[7] H. Biagioni, and, T. Gramchev, On smoothing phenomena for systems of evolution equations with dissipation, preprint. · Zbl 0990.35060
[8] Brezis, H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, (), 101-165
[9] Brezis, H., Problèmes unilateraux, J. math. pure appl., 51, 1-168, (1972) · Zbl 0237.35001
[10] Brezis, H.; Cazenáve, T., A nonlinear heat equation with singular initial data, J. anal. math., 68, 277-304, (1996) · Zbl 0868.35058
[11] Brezis, H.; Friedman, A., Nonlinear parabolic equations involving measures as initial conditions, J. math. pures appl., 62, 73-97, (1983) · Zbl 0527.35043
[12] Carvalho, A.; Oliva, S.M.; Pereira, A.L.; Rodriguez-Bernal, A., Attractors for parabolic problems with nonlinear boundary conditions, J. math. anal. appl., 207, 409-461, (1997) · Zbl 0876.35059
[13] Evans, L., Regularity properties of the heat equation subject to nonlinear boundary constraints, Nonlinear anal., 1, 593-602, (1977) · Zbl 0369.35034
[14] Henry, D., Geometric theory of semilinear parabolic equations, Lecture notes in mathematics, 840, (1981), Springer-Verlag Berlin
[15] Kozono, H.; Yamazaki, M., Semilinear heat equations and the navier – stokes equation with distributions in new function spaces as initial data, Comm. partial differential equations, 19, 959-1014, (1994) · Zbl 0803.35068
[16] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Basel · Zbl 0816.35001
[17] Moser, J., A sharp form of an inequaliy by N. Trudinger, Indiana U. math. J, 20, 1077-1092, (1971) · Zbl 0213.13001
[18] Tribel, H., Interpolation theory, function spaces, differential operators, (1978), North Holland Amsterdam
[19] Trudinger, N.S., On imbeddings into Orlicz spaces and some applications, J. math. mech., 17, 473-483, (1967) · Zbl 0163.36402
[20] Weisler, F.B., Semilinear evolution equations in Banach spaces, J. functional anal., 32, 277-296, (1979) · Zbl 0419.47031
[21] Weisler, F.B., Local existence and nonexistence for semilinear parabolic equations in Lp, Indiana U. math. J., 29, 79-102, (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.