In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax L-A pair. Using it we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and may even include functions which tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schr{ö}dinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.
We establish the nonlinear stability of $N$-soliton solutions of the modified Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of $1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on the variational characterization of $N$-solitons. We show that the $N$-solitons realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to fixed constraints on the $N$ first conserved quantities.To this aim, we construct a functional for which $N$-solitons are critical points, we prove…
We prove local existence and uniqueness of solutions of the focusing modified Korteweg - de Vries equation $u_t + u^2u_x + u_{xxx} = 0$ in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of $x$. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to $x$ and that solves a generalized version of the focusing mKdV equation. The latter equation is solved by discretization methods.
The present paper is devoted to weighted Nonlinear Schrödinger- Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators $\mathcal{L}$ in $\mathbb{R}^N$ and we establish some criteria for an unbounded open set to be a Maximum Principle set for $\mathcal{L}$. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author).
This paper deals with a class of singularly perturbed nonlinear elliptic problems $(P_\e)$ with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as $\e\to 0$, and the domain is supposed to be unbounded and with unbounded boundary. Domains that enlarge at infinity, and whose boundary flattens or shrinks at infinity, are considered. It is proved that in such domains problem $(P_\e)$ has at least 2 solutions.
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers co…
We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the $n$-dimensional whole space and discuss the energy identity for solutions in $L^q$. As a co…
We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there …
We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan-Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also…
We derive asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g., various paraboloids and horns.
We investigate a version of the Phragmén-Lindelöf theorem for solutions of the equation $Δ_\infty u=0$ in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the $p$-harmonic equation when $p$ tends to $\infty$.
We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space $R^n_+, n \ge 2$, is not subadditive on null sets when $p \neq 2$. Using their result and spherical inversion, we create similar…
This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)ΔU(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive. Under the behavior of the diffusion coefficient $D$ with polynomial order at spatial infinity, a family of supersolutions with slowly decaying property at spatial infinity is provided. As a first application, weighted $L^2$ type decay estimates for the initial-boundary value problem of the corresponding parabolic equation are proved. The second application is the …
We study positive bound states for the semiclassical stationary nonlinear Schrödinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, the potentials might be singular at the origin or vanish superquadratically at infinity.
We consider Leray's problem on stationary Navier-Stokes flows with arbitrary large fluxes in an unbounded cylinder with several exits to infinity. For a stationary Navier-Stokes flow with large fluxes in the unbounded cylinder we prove that, if the difference between the pressure of the main flow and the pressure of the Poiseuille flow with the same flux in a branch of the cylinder satisfies some asymptotic boundedness condition at $|x|\rightarrow \infty$ then the flow behaves at infinity of the branch like the Poiseuille flow.
We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip subject to a new deformation of "shearing": the strip is built by translating a segment oriented in a constant direction along an unbounded curve in the plane. We locate the essential spectrum under the hypothesis that the projection of the tangent vector of the curve to the direction of the segment admits a (possibly unbounded) limit at infinity and state sufficient conditions which guarantee the existence of discrete eigenvalues. We justify the optimality of these conditions by establishing a spectral stability in op…
We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }Ω\subseteq \mathbb{R}^{N},\ N\geq 3, \] where $Ω$ is a radial domain (bounded or unbounded) and $u$ satisfies $u=0$ on $\partial Ω$ if $Ω\neq \mathbb{R}^{N}$ and $u\rightarrow 0$ as $\left| x\right| \rightarrow \infty $ if $Ω$ is unbounded. The potential $V$ may be vanishing or unbounded at zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear. If $g$ is sublinear, …
