Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

In this paper we study the Dirichlet problem

$\left\{\begin{array}{lll}-\Delta_p{u} = \sigma |u|^{p-2}u + \omega \quad {\rm in}\;\Omega,\\ u = 0 \qquad\quad\qquad\quad\;\qquad{\rm on}\;\partial\Omega,\end{array}\right.$

, where σ and ω are nonnegative Borel measures, and \({\Delta_p{u} = \nabla \cdot (\nabla{u} \, |\nabla{u}|^{p-2})}\) is the p-Laplacian. Here \({\Omega \subseteq \mathbf{R}^n}\) is either a bounded domain, or the entire space. Our main estimates concern optimal pointwise bounds of solutions in terms of two local Wolff’s potentials, under minimal regularity assumed on σ and ω. In addition, analogous results for equations modeled by the k-Hessian in place of the p-Laplacian will be discussed.

     

Large Time Behavior for a Quasilinear Diffusion Equation with Critical Gradient Absorption

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption

$$\begin{aligned} \partial _t u-\Delta _{p}u+|\nabla u|^{q_*}=0 \quad \hbox {in}\, (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

for \(p\in (2,\infty )\) and \(q_*:=p-N/(N+1)\). We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the p-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.

     

Convergence of Global and Bounded Solutions of Some Nonautonomous Second Order Evolution Equations with Nonlinear Dissipation

In this paper we study the asymptotic behavior of solutions of the following nonautonomous wave equation with nonlinear dissipation.

$\left\{\begin{array}{ll} u_{tt}+\vert u_{t}\vert^{\alpha}u_{t}-\Delta u +f(u)=g(t,x),\quad{\rm in}\,\mathbb{R}_{+}\times\Omega,\\ \qquad\qquad u(t,x)=0,\quad\, {\rm on}\,\mathbb{R}_{+}\times\partial\Omega,\end{array}\right.$

where f is an analytic function, α is a small positive real and g(t, ·) tends to 0 sufficiently fast in L ²(Ω) as t tends to ∞.

We also obtain a general convergence result and the rate of decay of solutions for a class of second order ODE containing as a special case

$\left\{\begin{array}{ll} \ddot{U}(t)+\Vert\dot{U}(t)\Vert^{\alpha}\dot{U}(t)+\nabla F(U(t))=g(t),\quad t \in \mathbb{R}_+,\\ \qquad U(0)=U_{0}\,\in \mathbb{R}^{N},\quad\dot{U}(0)=U_{1}\in \mathbb{R}^{N}. \end{array}\right.$

     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

The Airy Function is a Fredholm Determinant

Let G be the Green’s function for the Airy operator

$$\begin{aligned} L\varphi := -\varphi ''+ x \varphi , \quad 0< x < \infty , \quad \varphi (0)=0. \end{aligned}$$

We show that the integral operator defined by G is Hilbert–Schmidt and that the 2-modified Fredholm determinant

$$\begin{aligned} {\mathrm {det}}_2(1+zG) = \frac{{\mathrm {Ai}}(z)}{{\mathrm {Ai}}(0)} , \quad z \in {\mathbb {C}}. \end{aligned}$$

     

Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions

In this paper we establish, using variational methods, the existence and multiplicity of weak solutions for a general class of quasilinear problems involving \(p(\cdot )\)-Laplace type operators, with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz (A-R) type growth conditions, namely

$$\begin{aligned} \left\{ \begin{array}{rcll} -{\text {div}}(a(|\nabla u|^{p(x)})|\nabla u|^{p(x)-2}\nabla u)&{}=&{}\lambda f(x,u) &{} \text{ in } \Omega ,\\ u&{}=&{}0 &{} \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$

By different types of versions of the Mountain Pass Theorem with Cerami condition, as well as, the Fountain and Dual Theorem with Cerami condition, we obtain some existence of weak solutions for the above problem under some considerations. Moreover, we show that the problem treated has at least one nontrivial solution for any parameter \(\lambda >0\) small enough, and also that the solution blows up, in the Sobolev norm, as \(\lambda \rightarrow 0^{+}.\) Finally, by imposing additional hypotheses on the nonlinearity \(f(x,\cdot ),\) we get the existence of infinitely many weak solutions by using the Genus Theory introduced by Krasnoselskii.

     

Boundedness for Some Doubly Nonlinear Parabolic Equations in Measure Spaces

In the context of measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality, we derive local and global sup bounds of the nonnegative weak subsolutions of

$$\begin{aligned} (u^{q})_t-\nabla \cdot {(|\nabla u|^{p-2}\nabla u)}=0, \quad \mathrm {in} \ U_\tau = U \times (\tau _1, \tau _2] , \quad p>1,\quad q>1 \end{aligned}$$

and of its associated Dirichlet problem, respectively. For particular ranges of the exponents p and q, we show that any locally nonnegative weak subsolution, taken in \(Q (\subset \bar{Q}\subset U_\tau )\), is controlled from above by the \(L^\alpha (\bar{Q}) \)-norm, for \(\alpha = \max \{p, q+1\}\). As for the global setting, under suitable assumptions on the boundary datum g and on the initial datum, we obtain sup bounds for u, in \(U \times \{ t\}\), which depend on the \(\sup g\) and on the \(L^{q+1}(U \times (\tau _1, \tau _1+t])\)-norm of \((u-\sup g)_+\), for all \(t \in (0, \tau _2-\tau _1]\). On the critical ranges of p and q, a priori local and global \(L^\infty \) estimates require extra qualitative information on u.

     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and f, g smooth in all variables.

     

Gradient Bounds for Elliptic Problems Singular at the Boundary

Let Ω be a bounded smooth domain in \({{R}^N, N \geqq 2}\), and let us denote by d(x) the distance function d(x, ?Ω). We study a class of singular Hamilton–Jacobi equations, arising from stochastic control problems, whose simplest model is

$ - \alpha \Delta u+ u + \frac{\nabla u \cdot B (x)}{d (x)}+ c(x) |\nabla u|^2=f (x) \quad {\rm in}\,\Omega, $

where f belongs to \({W^{1,\infty}_{\rm loc} (\Omega)}\) and is (possibly) singular at \({\partial \Omega, c\in W^{1,\infty} (\Omega)}\) (with no sign condition) and the field \({B\in W^{1,\infty} (\Omega)^N}\) has an outward direction and satisfies \({B\cdot \nu\geqq \alpha}\) at ?Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.

     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Invariant Tori of Full Dimension for Second KdV Equations with the External Parameters

In this paper, we consider the second KdV equation with the external parameters

$$\begin{aligned} u_{t} =\partial _x^5 u +(M_{\sigma }u+u^3)_{x}, \end{aligned}$$

under zero mean-value periodic boundary conditions

$$\begin{aligned} u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

where \(M_\sigma \) is a real Fourier multiplier. It is proved that the equations admit a Whitney smooth family of small amplitude, real analytic almost periodic solutions with all frequencies. The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property of the perturbation and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property of the perturbation, our normal form part is independent of angle variables in spite of the unbounded perturbation. This is the first attempt to prove the almost periodic solutions for the unbounded perturbation case.

     

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\):

$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$

where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.

     

Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.     

Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

We study the Neumann boundary value problem for the second order ODE

$$\begin{aligned} u^{\prime \prime } + (a^+(t)-\mu a^-(t))g(u) = 0, \qquad t \in [0,T], \end{aligned}$$

(1)

where \(g \in {\mathcal {C}}^1({\mathbb {R}})\) is a bounded function of constant sign, \(a^+,a^-: [0,T] \rightarrow {\mathbb {R}}^+\) are the positive/negative part of a sign-changing weight \(a(t)\) and \(\mu > 0\) is a real parameter. Depending on the sign of \(g^{\prime }(u)\) at infinity, we find existence/multiplicity of solutions for \(\mu \) in a “small” interval near the value

$$\begin{aligned} \mu _c = \frac{\int _0^T a^+(t) \, dt}{\int _0^T a^-(t) \, dt}\,. \end{aligned}$$

The proof exploits a change of variables, transforming the sign-indefinite Eq. (1) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for \(\mu \rightarrow 0^+\) and \(\mu \rightarrow +\infty \) are given, as well.

     

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

This paper deals with an initial-boundary value problem for the system $$\left\{ \begin{array}{llll} n_t + u\cdot\nabla n &=& \Delta n -\nabla \cdot (n\chi(c)\nabla c), \quad\quad & x\in\Omega, \, t > 0,\\ c_t + u\cdot\nabla c &=& \Delta c-nf(c), \quad\quad & x\in\Omega, \, t > 0,\\ u_t + \kappa (u\cdot \nabla) u &=& \Delta u + \nabla P + n \nabla\phi, \qquad & x\in\Omega, \, t > 0,\\ \nabla \cdot u &=& 0, \qquad & x\in\Omega, \, t > 0,\end{array} \right.$$ which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains ${\Omega \subset \mathbb{R}^2}$ and under appropriate assumptions on the parameter functions χ, f and ?, for each ${\kappa\in\mathbb{R}}$ and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium ${(\overline{n_0},0,0)}$ , where ${\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}$ , in the sense that as t→∞, $$n(\cdot,t) \to \overline{n_0}, \qquad c(\cdot,t) \to 0 \qquad \text{and}\qquad u(\cdot,t) \to 0$$ hold with respect to the norm in ${L^\infty(\Omega)}$ .     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force

$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$

where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.

     

Asymptotic Behavior of Type I Blowup Solutions to a Parabolic-Elliptic System of Drift–Diffusion Type

We consider a Cauchy problem for a parabolic-elliptic system of drift–diffusion type. The problem is formally of the form

$ U_t = \nabla \cdot (\nabla U-U \nabla (-\Delta)^{-1}U). $

This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I, in the sense that its blowup rate is the same as the corresponding ordinary differential equation y _t  = y ² (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar, provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three.