Abstract: We show that the uniform radius of spatial analyticity \(\sigma (t)\) of solutions at time t to the cubic nonlinear Schrödinger equations (NLS) on the circle cannot decay faster than 1 / t as \( t \rightarrow \infty \) , given initial data that is analytic with fixed radius \(\sigma _0\) . The same decay rate has been recently established for cubic NLS on the real line. PubDate: 2019-03-21
Abstract: We are concerned with the existence of solutions for the magnetohydrodynamics with power-law type nonlinear viscous fluid in \({{\mathbb {R}}}^3\) . We construct weak solutions for \(q>8/5\) , and furthermore, in case \(q\ge 5/2\) , we prove the existence of strong solutions. PubDate: 2019-03-18
Abstract: The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly non-unique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones. PubDate: 2019-02-25
Abstract: Andrews and Bryan (J Reine Angew Math 653:179–187, 2011) discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of the arc length between its endpoints and elapsed time. They apply this estimate to short segments and deduce directly that the maximum curvature decays exponentially to the curvature of a circle with the same length. We consider the expansion of convex curves under inverse (mean) curvature flow and show that the above comparison function also works in this case to obtain a new proof of the fact that the flow exists for all times and becomes round in shape, i.e. converges smoothly to the unit circle after an appropriate rescaling. PubDate: 2019-02-18
Abstract: In this paper we shall study the movable singularity of semi linear Heun equation and its application to the blowup of a semi linear wave equation. In fact, the semi linear Heun equation appears if we consider a radially symmetric self-similar solution of the semi linear wave equation. By the movable singularity we mean the singularity which does not appear in the coefficients of the equation and that depends on the respective solution. We focus on movable singularity when we construct a singular solution of the semi linear wave equation with singularities on the characteristic cone. In the proof of our theorem we reduce our equation to a simpler form by the method similar to the so-called Birkhoff reduction, then we analyze the reduced equation. The latter part is closely related with the parametrization of a solution in terms of the Jacobi elliptic function. PubDate: 2019-02-05
Abstract: In this paper we investigate the existence and uniqueness of weak solutions of the nonautonomous Hamilton–Jacobi–Bellman equation on the domain \((0,\infty ) \times \Omega \) . The Hamiltonian is assumed to be merely measurable in time variable and the open set \(\Omega \) may be unbounded with nonsmooth boundary. The set \(\overline{\Omega }\) is called here a state constraint. When state constraints arise, then classical analysis of Hamilton–Jacobi–Bellman equation lacks appropriate notion of solution because continuous solutions could not exist. In this work we propose a notion of weak solution for which, under a suitable controllability assumption, existence and uniqueness theorems are valid in the class of lower semicontinuous functions vanishing at infinity. PubDate: 2019-02-04
Abstract: We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared in the pointwise sense to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup. PubDate: 2019-02-01
Abstract: We prove generalized semiconcavity results, jointly in time and state variables, for the value function of a stochastic finite horizon optimal control problem, where the evolution of the state variable is described by a general stochastic differential equation (SDE) of jump type. Assuming that terms comprising the SDE are \(C^1\) -smooth, and that running and terminal costs are semiconcave in generalized sense, we show that the value function is also semiconcave in generalized sense, estimating the semiconcavity modulus of the value function in terms of smoothness and generalized semiconcavity moduli of data. Of course, these translate into analogous regularity results for (viscosity) solutions of integro-differential Hamilton–Jacobi–Bellman equations due to their controllistic interpretation. This paper may be seen as a sequel to Feleqi (Dyn Games Appl 3(4):523–536, 2013), where we dealt with the generalized semiconcavity of the value function only in the state variable. PubDate: 2019-01-17
Abstract: The purpose of this short note is to attract attention to the concept of the upper perturbation property of \(L^n\) -viscosity subsolutions introduced in Crandall et al. (in: On the equivalence of various weak notions of solutions of elliptic PDEs with measurable ingredients, progress in elliptic and parabolic partial differential equations (Capri, 1994), Longman, Harlow, 1996). We show that a recent result of Braga and Moreira (NoDEA Nonlinear Differ Equ Appl 25(2):12, 2018) about removable sets for viscosity solutions of fully nonlinear degenerate elliptic PDE is an easy consequence of the upper perturbation property. We also prove a parabolic result about removable sets. PubDate: 2019-01-03
Abstract: We study a class of viscous, incompressible non-Newtonian fluids in two space dimensions with periodic boundary conditions and an additive Gaussian noise. The nonlinear elliptic operator related to the stress tensor possesses p-structure. When the fluid is shear thickening, we prove that the associated Kolmogorov operator is essentially m-dissipative in a space with respect to an invariant measure. In addition, if the viscosity constant is sufficiently large, we show that the invariant measure is exponential mixing so that it is also unique. PubDate: 2019-01-02
Abstract: In this paper, we study the intermittent property for the following nonlinear stochastic partial differential equation (SPDE in the sequel) in (1+1)-dimension $$\begin{aligned} \left( \frac{\partial }{\partial t}+q(x,D_x)\right) u(t,x)= g(u(t,x))\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x),\quad t>0 \quad \mathrm{and} \quad x\in {\mathbb {R}}, \end{aligned}$$ with \(q(x,D_x)\) is a pseudo-differential operator which generates a stable-like process. The forcing noise denoted by \(\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x)\) is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet \(w_\rho (t,x)\) , we prove that the solution is weakly full intermittent based on the moment estimates of the solution. PubDate: 2019-01-02
Abstract: We consider the Cauchy problem of the semilinear wave equation with a damping term $$\begin{aligned} \left\{ \begin{array}{ll} u_{tt} - \Delta u + c(t,x) u_t = u ^p,&{}(t,x)\in (0,\infty )\times {\mathbb {R}}^N,\\ u(0,x) = \varepsilon u_0(x), \quad u_t(0,x) = \varepsilon u_1(x),&{} x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$ where \(p>1\) and the coefficient of the damping term has the form $$\begin{aligned} c(t,x) = a_0 (1+ x ^2)^{-\alpha /2} (1+t)^{-\beta } \end{aligned}$$ with some \(a_0 > 0\) , \(\alpha < 0\) , \(\beta \in (-1, 1]\) . In particular, we mainly consider the cases $$\begin{aligned} \alpha< 0, \beta =0 \quad \text{ or } \quad \alpha < 0, \beta = 1, \end{aligned}$$ which imply \(\alpha + \beta < 1\) , namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by $$\begin{aligned} p = 1+ \frac{2}{N-\alpha }. \end{aligned}$$ This shows that the critical exponent is the same as that of the corresponding parabolic equation $$\begin{aligned} c(t,x) v_t - \Delta v = v ^p. \end{aligned}$$ The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli–Kohn–Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima [15]. We also give an upper estimate of the lifespan. PubDate: 2018-11-20
Abstract: We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, \(u''+h(x,\lambda )u^p=0\) , \(u>0\) in \((-1,1)\) with \(u(-1)=u(1)=0\) , where \(p>1\) , \(h(x,\lambda )=0\) for \( x <\lambda \) and \(h(x,\lambda )=1\) for \(\lambda \le x \le 1\) and \(\lambda \in (0,1)\) is a bifurcation parameter. We shall show that the problem has a unique even positive solution \(U(x,\lambda )\) for each \(\lambda \in (0,1)\) . We shall prove that there exists a unique \(\lambda _*\in (0,1)\) such that a non-even positive solution bifurcates at \(\lambda _*\) from the curve \((\lambda , U(x,\lambda ))\) , where \(\lambda _*\) is explicitly represented as a function of p. PubDate: 2018-11-17
Abstract: This paper focuses on mathematical and numerical approaches to dynamic frictionless contact of nonlinear viscoelastic springs. This contact model is formulated by a nonlinear ordinary differential equation system and a pair of complementarity conditions. We propose three different numerical schemes in which each of them consists of several numerical methods. As a result, three groups of time-discrete numerical formulations are established. We use the coefficient of restitution to prove convergence of numerical trajectories, passing to the limit in the time step size. The Banach-fixed point theorem is applied to show the existence of global solutions satisfying all conditions. A new form of energy balance is derived, which is verified theoretically and numerically. All of the three schemes are implemented and their numerical results are compared with each other. PubDate: 2018-11-12
Abstract: In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p-Laplacian in the whole \({\mathbb {R}}^n\) . PubDate: 2018-10-19
Abstract: We consider the initial value problem for a fully-nonlinear degenerate parabolic equation with a dynamic boundary condition in a half space. Our setting includes geometric equations with singularity such as the level-set mean curvature flow equation. We establish a comparison principle for a viscosity sub- and supersolution. We also prove existence of solutions and Lipschitz regularity of the unique solution. Moreover, relation to other types of boundary conditions is investigated by studying the asymptotic behavior of the solution with respect to a coefficient of the dynamic boundary condition. PubDate: 2018-10-13
Abstract: We study a class of weakly coupled systems of Hamilton–Jacobi equations at the critical level. We associate to it a family of scalar discounted equation. Using control-theoretic techniques we construct an algorithm which allows obtaining a critical solution to the system as limit of a monotonic sequence of subsolutions. We moreover get a characterization of isolated points of the Aubry set and establish semiconcavity properties for critical subsolutions. PubDate: 2018-10-01
Abstract: We consider the Navier–Stokes equations in vorticity form in \(\mathbb {R}^2\) with a white noise forcing term of multiplicative type, whose spatial covariance is not regular enough to apply the Itô calculus in \(L^q\) spaces, \(1<q<\infty \) . We prove the existence of a unique strong (in the probability sense) solution. PubDate: 2018-09-28
Abstract: In this paper we study a phase transition model for vehicular traffic flows. Two phases are taken into account, according to whether the traffic is light or heavy. We assume that the two phases have a non-empty intersection, the so called metastable phase. The model is given by the Lighthill–Whitham–Richards model in the free-flow phase and by the Aw–Rascle–Zhang model in the congested phase. In particular, we study the existence of solutions to Cauchy problems satisfying a local point constraint on the density flux. We prove that if the constraint F is higher than the minimal flux \(f_\mathrm{c}^-\) of the metastable phase, then constrained Cauchy problems with initial data of bounded total variation admit globally defined solutions. We also provide sufficient conditions on the initial data that guarantee the global existence of solutions also in the case \(F < f_\mathrm{c}^-\) . These results are obtained by applying the wave-front tracking technique. PubDate: 2018-09-18
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