## Convergence to a steady state for asymptotically autonomous semilinear heat equations on $$\mathbb R^N$$.(English)Zbl 1263.35035

The authors treat semilinear parabolic equations of the form $u_t=\Delta u+f(u)+h(x,t),\qquad(x,t)\in\mathbb{R}^N\times(0,\infty), \tag{1}$ that are asymptotically autonomous in the sense that $$\|h(\cdot,t)\|_\infty\to0$$ as $$t\to\infty$$. Here $$f$$ is assumed to be continuously differentiable and to satisfy $$f(0)=0$$ and $$f'(0)<0$$. Consider a global, bounded and nonnegative solution $$u$$ of (1) that decays in $$x$$, uniformly in $$t$$: $\lim_{|x|\to\infty}\sup_{t>0}u(x,t)=0.$ Define its $$\omega$$-limit set in a suitable Banach space of states by $\omega(u):=\{v: u(\cdot,t_k)\to v\text{ for some sequence } t_k\to\infty\}.$ A positive solution of $$\Delta u+f(u)=0$$ on $$\mathbb{R}^N$$ that decays to $$0$$ as $$|x|\to\infty$$ is called a ground state. A ground state is always radially symmetric about some point and radially decaying.
Under these assumptions it is shown that either $$\omega(u)=\{0\}$$ or that $$\omega(u)$$ consists entirely of ground states. If in addition $$h(\cdot,t)$$ decays exponentially in a suitable Hölder class, then either $$\omega(u)=\{0\}$$ or $$\omega(u)$$ consists of exactly one ground state. Since $\lim_{t\to\infty}\text{dist}_\infty(u(\cdot,t),\omega(u))=0,$ these results yield, respectively, quasiconvergence and convergence of $$u$$ in the $$L^\infty$$-sense.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K58 Semilinear parabolic equations 35B09 Positive solutions to PDEs 35B07 Axially symmetric solutions to PDEs

### Keywords:

 [1] Aulbach, B., Continuous and discrete time dynamics near manifolds of equilibria, (1984), Springer-Verlag Berlin, Heidelberg · Zbl 0535.34002 [2] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. existence of a ground state, Arch. ration. mech. anal., 82, 313-345, (1983) · Zbl 0533.35029 [3] Brunovský, P.; Poláčik, P., On the local structure of ω-limit sets of maps, Z. angew. math. phys., 48, 976-986, (1997) · Zbl 0889.34048 [4] Busca, J.; Jendoubi, M.-A.; Poláčik, P., Convergence to equilibrium for semilinear parabolic problems in $$\mathbb{R}^N$$, Comm. partial differential equations, 27, 1793-1814, (2002) · Zbl 1021.35013 [5] Chen, C.-C.; Lin, C.-S., Uniqueness of the ground state solutions of $$\operatorname{\Delta} u + f(u) = 0$$ in $$\operatorname{R}^n$$, $$n \geqslant 3$$, Comm. partial differential equations, 16, 1549-1572, (1991) · Zbl 0753.35034 [6] Chen, X.-Y.; Poláčik, P., Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, J. dynam. differential equations, 7, 73-107, (1995) · Zbl 0822.35073 [7] Chill, R., On the łojasiewicz-Simon gradient inequality, J. funct. anal., 201, 2, 572-601, (2003) · Zbl 1036.26015 [8] Chill, R.; Haraux, A.; Jendoubi, M.A., Applications of the łojasiewicz-Simon gradient inequality to gradient-like evolution equations, Anal. appl. (singap.), 7, 4, 351-372, (2009) · Zbl 1192.34068 [9] Chill, R.; Jendoubi, M.A., Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear anal., 53, 1017-1039, (2003) · Zbl 1033.34066 [10] Chill, R.; Jendoubi, M.A., Convergence to steady states of solutions of non-autonomous heat equations in $$\mathbb{R}^N$$, J. dynam. differential equations, 19, 3, 777-788, (2007) · Zbl 1166.35005 [11] Cortázar, C.; del Pino, M.; Elgueta, M., The problem of uniqueness of the limit in a semilinear heat equation, Comm. partial differential equations, 24, 2147-2172, (1999) · Zbl 0940.35107 [12] Du, Y.; Matano, H., Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. eur. math. soc. (JEMS), 12, 279-312, (2010) · Zbl 1207.35061 [13] Fašangová, E., Asymptotic analysis for a nonlinear parabolic equation on $$\mathbb{R}$$, Comment. math. univ. carolin., 39, 525-544, (1998) · Zbl 0963.35080 [14] Fašangová, E.; Feireisl, E., The long-time behavior of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions, Proc. roy. soc. Edinburgh sect. A, 129, 319-329, (1999) · Zbl 0933.35101 [15] Feireisl, E.; Petzeltová, H., Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential integral equations, 10, 181-196, (1997) · Zbl 0879.35023 [16] Feireisl, E.; Poláčik, P., Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on R, Adv. differential equations, 5, 583-622, (2000) · Zbl 0987.35079 [17] Flores, J.G., On a threshold of codimension 1 for the Nagumo equation, Comm. partial differential equations, 13, 1235-1263, (1988) · Zbl 0665.35034 [18] Földes, J., Symmetry of positive solutions of asymptotically symmetric parabolic problems on $$\mathbb{R}^N$$, J. dynam. differential equations, 23, 45-69, (2011) · Zbl 1222.35013 [19] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in $$\mathbb{R}^n$$, (), 369-402 [20] Hale, J.K.; Raugel, G., Convergence in gradient-like systems with applications to PDE, J. appl. math. phys. (ZAMP), 43, 63-124, (1992) · Zbl 0751.58033 [21] Hempel, R.; Voigt, J., The spectrum of Schrödinger operators in $$L_p(\mathbb{R}^d)$$ and in $$C_0(\mathbb{R}^d)$$, (), 63-72 · Zbl 0822.47002 [22] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer-Verlag New York · Zbl 0456.35001 [23] Huang, S.-Z.; Takáč, P., Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear anal., 46, 5, 675-698, (2001) · Zbl 1002.35022 [24] Hurley, M., Chain recurrence, semiflows, and gradients, J. dynam. differential equations, 7, 437-456, (1995) · Zbl 0832.34041 [25] Jones, C., Spherically symmetric solutions of a reaction-diffusion equation, J. differential equations, 49, 142-169, (1983), MR704268 (84h:35084) · Zbl 0523.35059 [26] Kwong, M.K., Uniqueness of positive solutions of $$\operatorname{\Delta} u - u + u^p = 0$$ in $$\operatorname{R}^n$$, Arch. ration. mech. anal., 105, 243-266, (1989) · Zbl 0676.35032 [27] Ladyzhenskaya, O.A.; Solonnikov, V.A.; Urallʼceva, N.N., Linear and quasilinear equations of parabolic type, Transl. math. monogr., vol. 23, (1967), Nauka Moscow, Russian original: · Zbl 0164.12302 [28] Li, C., Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. partial differential equations, 16, 585-615, (1991) · Zbl 0741.35014 [29] Li, Y.; Ni, W.-M., Radial symmetry of positive solutions of nonlinear elliptic equations in $$\operatorname{R}^n$$, Comm. partial differential equations, 18, 1043-1054, (1993) · Zbl 0788.35042 [30] Lieberman, G.M., Second order parabolic differential equations, (1996), World Scientific Publishing Co. Inc. River Edge, NJ · Zbl 0884.35001 [31] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Berlin · Zbl 0816.35001 [32] Mischaikow, K.; Smith, H.; Thieme, H.R., Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. amer. math. soc., 347, 5, 1669-1685, (1995) · Zbl 0829.34037 [33] Poláčik, P., Symmetry properties of positive solutions of parabolic equations on $$\mathbb{R}^N$$: II. entire solutions, Comm. partial differential equations, 31, 1615-1638, (2006) · Zbl 1128.35051 [34] Poláčik, P., Threshold solutions and sharp transitions for nonautonomous parabolic equations on $$R^N$$, Arch. ration. mech. anal., 199, 69-97, (2011) · Zbl 1262.35130 [35] Simon, L., Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. math., 118, 525-571, (1983) · Zbl 0549.35071 [36] () [37] Zlatoš, A., Sharp transition between extinction and propagation of reaction, J. amer. math. soc., 19, 251-263, (2006) · Zbl 1081.35011