Nonlinear Differential Equations and Applications (NoDEA)最新文献

We prove some multiplicity results for Neumann-type boundary value problems associated with a Hamiltonian system. Such a system can be seen as the weak coupling of two systems, the first of which has some periodicity properties in the Hamiltonian function, the second one presenting the existence of a well-ordered pair of lower/upper solutions.

When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear acoustic equations of fractional higher order. In this work, we relate them to the classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxation times. To this end, we perform a singular limit analysis and determine their behavior as the relaxation time tends to zero. We show that, depending on the nonlinearities and assumptions on the data, these models can be seen as approximations of the Westervelt, Blackstock, or Kuznetsov wave equations in nonlinear acoustics. We furthermore establish the convergence rates and thus determine the error one makes when exchanging local and nonlocal models. The analysis rests upon the uniform bounds for the solutions of the acoustic equations with fractional higher-order derivatives, obtained through a testing procedure tailored to the coercivity property of the involved (weakly) singular memory kernel.

We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on ((x,u,nabla u)), and with a convective term f. The assumptions on the members of the equation are formulated in terms of Young’s functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.

We prove that on a compact Riemannian manifold of dimension 3 or higher, with positive Ricci curvature, the Allen–Cahn min–max scheme in Bellettini and Wickramasekera (The Inhomogeneous Allen–Cahn Equation and the Existence of Prescribed-Mean-Curvature Hypersurfaces, 2020), with prescribing function taken to be a non-zero constant (lambda ), produces an embedded hypersurface of constant mean curvature (lambda ) ((lambda )-CMC). More precisely, we prove that the interface arising from said min–max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of Bellettini and Wickramasekera, 2020). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed (lambda )-CMC hypersurfaces (with Morse index 1) for any prescribed non-zero constant (lambda ), with the expected singular set when the ambient dimension is 8 or higher.

Abstract

In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system $$begin{aligned} {left{ begin{array}{ll} -Delta u+V(x)u+ K(x)phi u=f(x,u),quad &{}xin {mathbb {R}}^3, -Delta phi -varepsilon ^4Delta _4phi = K(x) u^2, &{}xin {mathbb {R}}^3, end{array}right. } end{aligned}$$ where (varepsilon ) is a positive parameter and f is linearly bounded in u at infinity. Under suitable assumptions on V, K and f, we establish the existence and asymptotic behavior of ground state solutions to the system. We prove that they converge to the solutions of the classic Schrödinger–Poisson system associated as (varepsilon ) tends to zero.

We consider the total energy decay together with the (L^{2})-bound of the solution itself of the Cauchy problem for wave equations with a short-range potential and a localized damping, where we treat it in the one-dimensional Euclidean space (textbf{R}). To study these, we adopt a simple multiplier method. In this case, it is essential that compactness of the support of the initial data not be assumed. Since this problem is treated in the whole space, the Poincaré and Hardy inequalities are not available as have been developed for the exterior domain case with (n ge 1). However, the potential is effective for compensating for this lack of useful tools. As an application, the global existence of a small data solution for a semilinear problem is demonstrated.

Abstract

The asymptotic behavior of solutions as a small parameter tends to zero is determined for a variety of singular-limit PDEs. In some cases even existence for a time independent of the small parameter was not known previously. New examples for which uniform existence does not hold are also presented. Our methods include both an adaptation of geometric optics phase analysis to singular limits and an extension of that analysis in which the characteristic variety determinant condition is supplemented with a periodicity condition.

The aim of this article is to study partial regularity of a minimizer (varvec{u:Omega subset mathbb {R}^n rightarrow mathbb {R}^N}) for a double phase functional with variable exponents:

where (varvec{{ vert } cdot { vert }_A}) stands for the norm deduced from a positive definite sufficiently continuous tensor field (varvec{A:=big (A_{alpha beta }^{ij} (x,u)big )~~((x,u) in Omega times mathbb {R}^N)}). We show that a minimizer (varvec{u}) is in the class (varvec{C^{1,gamma }(Omega _1;mathbb {R}^N)}) for some constant (varvec{gamma in (0,1)}) and open subset (varvec{Omega _1 subset Omega }). We obtain also an estimate for the Hausdorff dimension of (varvec{Omega setminus Omega _1}).

In this paper we consider the problem of minimizing functionals of the form (E(u)=int _B f(x,nabla u) ,dx) in a suitably prepared class of incompressible, planar maps (u: B rightarrow mathbb {R}^2). Here, B is the unit disk and (f(x,xi )) is quadratic and convex in (xi ). It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global minimizer of E(u) provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional (f(x,xi )), depending smoothly on (xi ) but discontinuously on x, whose unique global minimizer is the so-called (N-)covering map, which is Lipschitz but not (C^1).

Abstract

In this paper, we derive suitable optimal (L^p-L^q) decay estimates, (1le ple 2le qle infty ) , for the solutions to the (sigma ) -evolution equation, (sigma >1) , with scale-invariant time-dependent damping and power nonlinearity  (|u|^p) , $$begin{aligned} u_{tt}+(-Delta )^sigma u + frac{mu }{1+t} u_t= |u|^{p}, quad tge 0, quad xin {{mathbb {R}}}^n, end{aligned}$$ where  (mu >0) , (p>1) . The critical exponent (p=p_c) for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly (mu in (0, 1)) or (mu >1) . Under the assumption of small initial data in (L^m({{mathbb {R}}}^n)cap L^2({{mathbb {R}}}^n), m=1,2) , we find the critical exponent at low space dimension n with respect to (sigma ) , namely, $$begin{aligned} p_c= max left{ {{bar{p}}}(gamma _{m}), {{bar{p}}} (gamma _{m}+mu -1) right} , quad gamma _{m}{mathrm {,:=,}}frac{n}{msigma }, quad mu >1-gamma _m, end{aligned}$$ where ( {{bar{p}}}(gamma ){mathrm {,:=,}}1+ frac{2}{gamma }) is the well known Fujita exponent. Hence, (p_c={{bar{p}}}(gamma _{m})) if (mu >1) , whereas (p_c={{bar{p}}} (gamma _{m}+mu -1)) is a shift of Fujita type exponent if (mu in (0, 1)) .