We consider the Cauchy problem u_t=Δ u+f(u), x∈ℝ^N, t>0, u(x,0)=u_0(x), x∈ℝ^N, where N≥ 2 , f is a C^1 function satisfying minor nondegeneracy conditions, and u_0 is a radially symmetric function having a finite limit ζ as |x|→∞ . We have previously proved that if ζ is a stable equilibrium of the equation ξ̇=f(ξ ) and the solution u is bounded, then u is quasiconvergent: its ω -limit set with respect to the topology of L_loc^∞ (ℝ^N) consists of steady states. In the present paper, we consider the case when ζ is linearly stable: f(ζ )=0 and f'(ζ )<0 . Under this condition, we show that if the solution of the above Cauchy problem is bounded, then it converges, locally uniformly with respect to x∈ℝ^N , to a single steady state.

We consider the Cauchy problem for the nonlinear heat equation u(t) = Delta u + f(u), x is an element of R-N, t > 0, where N >= 2 and f is a C-1 function satisfying minor nondegeneracy conditions. Our goal is to describe the large-time behavior of bounded solutions whose initial data are radially symmetric and have a finite limit zeta as vertical bar x vertical bar -> infinity. In the present paper, we examine the following two cases: f(zeta) not equal 0, or f(zeta) = 0 and zeta is a stable equilibrium of the equation (xi) over dot = f(xi). We prove that bounded solutions with such initial data are quasiconvergent: as t -> infinity, they approach a set of steady states in the topology of L-loc(infinity)(R-N).

We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.

We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution has finite limits $\theta^\pm$ as $x\to\pm\infty$, our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as $t\to\infty$ are steady states. It is known that this result is not valid in general if the limits are equal: $\theta^\pm=\theta_0$. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either $f(\theta_0)\ne0$, or $f(\theta_0)=0$ and $\theta_0$ is a stable equilibrium of the equation $\dot \xi=f(\xi)$. If $f(\theta_0)=0$ and $\theta_0$ is an unstable equilibrium of the equation $\dot \xi=f(\xi)$, we also prove some quasiconvergence theorem making (necessarily) additional assumptions on $u_0$. A major ingredient of our proofs of the quasiconvergence theorems---and a result of independent interest---is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.

We study positive partially localized solutions of the elliptic equation(1)Δxu+uyy+f(u)=0,(x,y)∈RN×R, where N≥2 and f is a C1 function satisfying f(0)=0 and f′(0)<0. By partially localized solutions we mean solutions u(x,y) which decay to zero as |x|→∞ uniformly in y. Our main concern is the existence of positive partially localized solutions which are quasiperiodic in y. The fact that such solutions can exist in equations of the above form was demonstrated in our earlier work: we proved that the nonlinearity f can be designed in such a way that equation (1) possesses positive partially localized quasiperiodic solutions with 2 frequencies. Our main contributions in the present paper are twofold. First, we improve the previous result by showing that positive partially localized quasiperiodic solutions with any prescribed number n≥2 of frequencies exist for some nonlinearities f. Second, we give a tangible sufficient condition on f which guarantees that equation (1) has such quasiperiodic solutions, possibly after f is perturbed slightly. The condition, with n=2, applies, for example, to some combined-powers nonlinearities f(u)=up+λuq−u with suitable exponents p>q>1 and coefficient λ>0.

We consider the equation 1 Δ _x u+u_yy+f(u)=0, x=(x_1,… ,x_N)∈ℝ^N, y∈ℝ, where N≥ 2 and f is a sufficiently smooth function satisfying f(0)=0 , f'(0)<0 , and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in x'=(x_1,… ,x_N-1) and decaying as |x'|→∞ , periodic in x_N , and quasiperiodic in y . Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type results.

We consider the semilinear heat equation $u_t=\Delta u+u^p$ on ${\mathbb R}^N$. Assuming that $N\ge 3$ and $p$ is greater than the Sobolev critical exponent $(N+2)/(N-2)$, we examine entire solutions (classical solutions defined for all $t\in {\mathbb R}$) and ancient solutions (classical solutions defined on $(-\infty,T)$ for some $T<\infty$). We prove a new Liouville-type theorem saying that if $p$ is greater than the Lepin exponent $p_L:=1+6/(N-10)$ ($p_L=\infty$ if $N\le 10$), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of radial symmetry; in other ranges of supercritical $p$ it is known not to be valid even in the class of radial solutions. Our other results include classification theorems for nonstationary entire solutions (when they exist) and ancient solutions, as well as some applications in the theory of blowup of solutions.

We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 0 are constant steady states, we investigate the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x, 0) are near γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f . The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In our proofs we employ phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x, t), ux(x, t)) : x ∈ R}, t > 0, of the solutions in question.

We consider a class of semilinear heat equations on R \mathbb {R} , including in particular the Fujita equation u t = u x x + | u | p − 1 u , x ∈ R , t ∈ R , \begin{equation*} u_t=u_{xx} +|u|^{p-1}u,\quad x\in \mathbb {R},\ t\in \mathbb {R}, \end{equation*} where p > 1 p>1 . We first give a simple proof and an extension of a Liouville theorem concerning entire solutions with finite zero number. Then we show that there is an infinite-dimensional set of entire solutions with infinite zero number.

In studies of superlinear parabolic equations ut=Δu+up,x∈RN,t>0,where p>1, backward self-similar solutions play an important role. These are solutions of the form u(x,t)=(T−t)−1∕(p−1)w(y), where y≔x∕T−t, T is a constant, and w is a solution of the equation Δw−y⋅∇w∕2−w∕(p−1)+wp=0. We consider (classical) positive radial solutions w of this equation. Denoting by pS, pJL, pL the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for p∈(pS,pJL) there are only countably many solutions, and for p∈(pJL,pL) there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.

Abstract We consider the Cauchy problem where f is a C1 function on with and u0 is a nonnegative continuous function on whose limits at are equal to 0. Assuming that the solution u is bounded, we study its asymptotic behavior as In the first part of this study, we proved a general quasiconvergence result: as the solution approaches a set of steady states in the topology of In this paper, we show that under certain generic, explicitly formulated conditions on the nonlinearity f, the solution necessarily converges to a single steady state in Then, under the same conditions, we describe the global asymptotic shape of the solution: the graph of has a top part close to the graph of and two sides taking shapes of “terraces” moving in the opposite directions with precisely determined speeds.

We consider the equation Delta u + u(yy) + f(u) = 0, (x, y) is an element of R-N x R, (1) where N >= 2 and f is a smooth function satisfying f (0) = 0 and f'(0) < 0. We show that for suitable nonlinearities f of this form equation (1) possesses uncountably many positive solutions which are quasiperiodic in y, radially symmetric in x, and decaying as vertical bar x vertical bar -> infinity uniformly in y. Our method is based on center manifold and KAM-type results and involves analysis of solutions of (1) in a vicinity of a y-independent solution u*(x)-a ground state of the equation Delta u + f (u) = 0 on R-N.

We consider the equation Delta u +u(yy) + f(x, u) = 0, (x, y) is an element of R-N x R (1) where f is sufficiently regular, radially symmetric in x, and f(., 0) 0. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in y and decaying as vertical bar x vertical bar -> infinity uniformly in y. Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of f(u) (x, 0) and f(uu) (x,0), and is independent of higher-order terms in the Taylor expansion of f(x, .). In particular, our results apply to some quadratic nonlinearities.

Abstract We consider a class of Schrödinger operators on ${\open R}^N$ with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in $L^2({\open R}^N)$, we show that the following properties are generic with respect to the potential: (P1) the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;(P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set. The genericity means that in suitable topologies the potentials having the above properties form a residual set. As we explain, (P1), (P2) are prerequisites for some applications of KAM-type results to nonlinear elliptic equations. Similar properties also play a role in optimal control and other problems in linear and nonlinear partial differential equations.

We consider the semilinear parabolic equation on the real line, where f is a locally Lipschitz function on We prove that if a solution u of this equation is bounded and its initial value has distinct limits at then the solution is quasiconvergent, that is, all its limit profiles as are steady states.

We consider semilinear parabolic equations u_t=u_xx+f(u) on ℝ . We give an overview of results on the large time behavior of bounded solutions, focusing in particular on their limit profiles as t→∞ with respect to the locally uniform convergence. The collection of such limit profiles, or, the ω -limit set of the solution, always contains a steady state. Questions of interest then are whether—or under what conditions—the ω -limit set consists of steady states, or even a single steady state. We give several theorems and examples pertinent to these questions.

We consider the equation $u_t=\Delta u+f(u)$ on $\mathbb{R}^N$. Under suitable conditions on $f$ and the initial value $u_0=u(\cdot,0)$, we show that as $t\to\infty$ the solution $u(\cdot,t)$ approaches a planar propagating terrace, or a stacked family of planar traveling fronts. Using this result, we show the asymptotic one-dimensional symmetry of $u(\cdot,t)$ as well as its quasi convergence in $L_{loc}^\infty(\mathbb{R}^N)$.