×

zbMATH — the first resource for mathematics

Convergence to steady states of solutions of non-autonomous heat equations in \(\mathbb R^N\). (English) Zbl 1166.35005
In this paper, the authors consider the following non-autonomous heat equation: \[ u_{t}-\Delta u+f(u)=g(t,x)\text{ for }(t,x)\in{\mathbb R}_{+}\times \mathbb{R}^{N},\quad u(0,x)=u_{0}(x)\text{ for }x\in\mathbb{R}^{N}\tag{1} \]
with \(N\geq3\) and \(f(u)=u-|u|^{p-1}u\) \((u\in\mathbb{R)}\) for some \(1<p<\frac{N+2}{N-2},g\in L^{2}(\mathbb{R}_{+};L^{2}\cap L^{q})\) for some \(q>N\) (where \(L^{p}:=L^{p}(\mathbb{R}^{N}),\) supp \(g(t,.)\subset K\) for all \(t\geq0\) and some \(K\subset\mathbb{R}^{N}\) compact, \(g(t,x)\geq0\) for all \((t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{N},\) and \(u_{0}\in H^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})\) has compact support.
The main result of the paper states that if \(u\in C(\mathbb{R}_{+};H^{1})\) is a positive solution of \((1)\) such that \(\sup_{t\in\mathbb{R}_{+}}\|u(t)\|_{H^{1}}<+\infty,\) and if there exists \(\delta>0\) such that \(\sup_{t\in\mathbb{R}_{+}}t^{1+\delta}\int_{t}^{\infty} \|g(s)\|_{L^{2}(\mathbb{R}^{N})}^{2}ds<+\infty,\) then \(\lim_{t\rightarrow\infty}u(t):=w\) exists in \(H^{1},\) and \(-\Delta w+f(w)=0.\)
This result extends previous results by C. Cortazar, M. del Pino and M. Elguetta [Commun. Partial Differ. Equations 24, No. 11–12, 2147–2172 (1999; Zbl 0940.35107)] and by E. Feireisl and H. Petzeltová [Differ. Integral Equ. 10, No. 1, 181–196 (1997; Zbl 0879.35023)] who considered the autonomous case \(g=0,\) as well as J. Busca, M. A. Jendoubi and P. Poláčik [Commun. Partial Differ. Equations 27, No. 9–10, 1793–1814 (2002; Zbl 1021.35013)] who considered a larger class of nonlinearities \(f.\)

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berestycki H., Lions P.L. (1983). Nonlinear scalar field equations I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–346 · Zbl 0533.35029
[2] Brézis H., (1992). Analyse fonctionnelle. Masson, Paris · Zbl 0511.46001
[3] Brunovský P., Poláčik P. (1997). On the local structure of {\(\omega\)}-limit sets of maps. Z. Angew. Math. Phys. 48, 976–986 · Zbl 0889.34048
[4] Busca J., Jendoubi M.A., Poláčik P. (2002). Convergence to equilibrium for semilinear parabolic problems in \(\mathbb{R}^{N}\) . Comm. Partial Differential Equations 27, 1793–1814 · Zbl 1021.35013
[5] Chen C.C., Lin C.S. (1991). Uniqueness of the ground state of solutions of {\(\Delta\)}u + f(u) = 0 in \(\mathbb{R}^{N}\) , n 3. Comm. Partial Differential Equations 16, 1549–1572 · Zbl 0753.35034
[6] Chill R. (2003). On the Łojasiewicz-Simon gradient inequality. J. Funct. Anal. 201, 572–601 · Zbl 1036.26015
[7] Chill R., Jendoubi M.A. (2003). Convergence to steady states in asymptotically autonomous semilinear evolution equations. Nonlinear Analysis, Ser. A: Theory Methods 53, 1017–1039 · Zbl 1033.34066
[8] Cortazar C., del Pino M., Elgueta M. (1999). The problem of uniqueness of the limit in a semilinear heat equation. Comm. Partial Differential Equations 24, 2147–2172 · Zbl 0940.35107
[9] Fašangová E. (1998). Asymptotic analysis for a nonlinear parabolic equation on \(\mathbb{R}\) . Math. Univ. Carolin. 39, 525–544 · Zbl 0963.35080
[10] Fašangová E., Feireisl E. (1999). The long-time behaviour of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions. Proc. Roy. Soc. Edinburgh 129A: 319–329 · Zbl 0933.35101
[11] Feireisl E., Petzeltová H. (1997). Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Differential Integral Equations 10, 181–196 · Zbl 0879.35023
[12] Haraux, A., and Jendoubi, M. A. (2002). On the convergence of global and bounded solutions of some evolution equations. Prépublication du Laboratoire Jacques-Louis Lions R02003 (2002). · Zbl 1145.35033
[13] Hille, E., and Phillips, R. S. (1957). Functional analysis and semi-groups. Amer. Math. Soc., Providence, R.I. · Zbl 0078.10004
[14] Huang S.-Z., Takáč P. (2001). Convergence in gradient-like systems which are asymptotically autonomous and analytic. Nonlinear Anal., Ser. A, Theory Methods 46, 675–698 · Zbl 1002.35022
[15] Kwong M.K. (1989). Uniqueness of positive solutions of {\(\Delta\)}u u + u p = 0 in \(\mathbb{R}^N\) . Arch. Rational Mech. Anal. 105, 243–266 · Zbl 0676.35032
[16] Lions, P.-L. (1988). On positive solutions of semilinear elliptic equations in unbounded domains, In Ni, W. M., Peletier, L. A., Serrin, J. (eds.), Nonlinear Diffusion Equations and Their Equilibrium States II Math. Sci. Res. Inst. Publ., Vol. 13, Springer Verlag, New York, Berlin, Heidelberg, 85–122.
[17] Ni W.M., Takagi I. (1993). Locating the peaks of least energy-solutions to a Neumann problem. Duke Math. J. 70, 247–281 · Zbl 0796.35056
[18] Simon L. (1996). Theorems on Regularity and Singularity of Energy Minimizing Maps. Lecture Notes in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. · Zbl 0864.58015
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.