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On the asymptotic behaviour of solutions of nonlinear parabolic equations. (English) Zbl 0538.35043

We study the asymptotic behaviour of classical solutions of nonlinear parabolic equations \[ -u_ t+a_{ij}(x,t,u,\nabla u)u_{x_ ix_ j}+f(x,t,u,\nabla u)=0\quad on\quad \Omega \times {\mathbb{R}}^+ \] with, say, \(u|_{\partial \Omega}=0\) and \(u(x,0)=u_ 0(x)\). Assuming w.l.o.g. that u converges to the trivial steady state we are interested in the exact decay rate for \(t\to \infty\). Under some natural conditions (e.g. quadratic growth of f(x,t,u,p) with respect to p for estimating the gradient) it is shown, that \[ | u(x,t)| +| u_ t(x,t)| +| \nabla u(x,t)| \leq Ce^{-\lambda_ 0t}\quad on\quad {\bar \Omega}\times {\mathbb{R}}^+ \] where \(\lambda_ 0>0\) is the first eigenvalue of the linearized equation \(-Lw=-A_{ij}w_{x_ ix_ j}- A_ iw_{x_ i}-A_ 0w,\) assuming \(|(\partial f/\partial u)(x,t,0,0)-A_ 0(x)| \in L_ 1({\mathbb{R}}^+), A_ 0\leq 0\), and similar conditions for the other terms. These estimates are sharp in general. The results are applied to the solutions of two reaction- diffusion systems.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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References:

[1] Amann, H.: Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations. Proc. Roy. Soc. Edinburgh,81A, 35-47 (1978) · Zbl 0406.35038
[2] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. Siam Review18, 620-709 (1976) · Zbl 0345.47044 · doi:10.1137/1018114
[3] Ewer, J.P.G.: On the asymptotic properties of a class of nonlinear parabolic equations. Applicable Anal.13, 249-260 (1982) · Zbl 0498.35012 · doi:10.1080/00036818208839396
[4] Friedman, A.: Partial differential equations of parabolic type., Englewood Cliffs, N.J.: Prentice-Hall 1964 · Zbl 0144.34903
[5] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations in second order. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0361.35003
[6] Lady?enskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. Providence, Rhode Island: American Math. Soc. (1968)
[7] Lieberman, G.M.: Interior gradient bounds for non-uniformly parabolic equations. Indiana Univ. Math. J.32, 579-601 (1983) · Zbl 0491.35021 · doi:10.1512/iumj.1983.32.32041
[8] Serrin, J.: Gradient estimates for solutions of nonlinear elliptic and parabolic equations in ?Contributions to Nonlinear Functional Analysis?. New York: Academic Press 565-602 (1971)
[9] Smoller, J.: Shock waves and reaction diffusion equations. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0508.35002
[10] Walter, W.: Differential and integral inequalities. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0252.35005
[11] Alikakos, N.D.:L p -bounds of solutions of reaction diffusion equation., Comm. Partial Differential Equations4, 827-868 (1979) · Zbl 0421.35009 · doi:10.1080/03605307908820113
[12] Bebernes, J.W., Chueh, K.N., Fulks, W.: Some applications of invariance for parabolic systems. Indiana Univ. Math. J.28, 269-277 (1979) · Zbl 0402.35056 · doi:10.1512/iumj.1979.28.28019
[13] Kahane, C.S.: On the asymptotic behaviour of solutions of a mildly nonlinear parabolic system. J. Differential Equations32, 454-471 (1979) · Zbl 0401.35060 · doi:10.1016/0022-0396(79)90044-5
[14] Gmira, A., Veron, L.: Asymptotic Behaviour of the Solution of a Semilinear Parabolic Equation. Monatshefte Math.94, 299-311 (1982) · Zbl 0502.35014 · doi:10.1007/BF01667384
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