Showing papers in "Journal of Evolution Equations in 2018"

Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components

Abstract: In a bounded domain $$\Omega \subset \mathbb {R}^2$$ , we consider the the chemotaxis-Stokes system $$\begin{aligned} \left\{ \begin{array}{ll} n_t + u\cdot abla n = \Delta n - abla \cdot \Big (nS(x,n,c) \cdot abla c \Big ), \qquad &{} x\in \Omega , \ t>0,\\ c_t + u\cdot abla c = \Delta c - nf(c), \qquad &{} x\in \Omega , \ t>0,\\ u_t = \Delta u + abla P + n abla \phi , \qquad abla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ which arises as a model for populations of aerobic bacteria swimming in a sessile water drop. In accordance with refined modeling approaches, we do not necessarily assume the chemotactic sensitivity S herein to be a scalar function, but rather allow S to attain values in $$\mathbb {R}^{2\times 2}$$ . As compared to the well-studied case of scalar-valued sensitivities in which an analysis can be based on favorable energy-type inequalities, this modification brings about significant new challenges which require to adequately cope with only little a priori information on regularity of solutions of ( $$\star $$ ). The present work creates a functional setup which despite this allows for the construction of certain global mass-preserving generalized solutions to an associated initial-boundary value problem in planar convex domains with smooth boundary, provided that the initial data and the parameter functions S, f and $$\phi $$ are sufficiently smooth, and that S is bounded and f is nonnegative with $$f(0)=0$$ .

On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities

Abstract: This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrodinger equation with combined power-type nonlinearities $$\begin{aligned} iu_{t}+\Delta u=\lambda _1|u|^{p_1}u+\lambda _2|u|^{p_2}u. \end{aligned}$$ When $$p_1=\frac{4}{N}$$ and $$0

Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term

Abstract: In this paper, we consider a plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping. Using the Galaerkin method, we establish the existence of solutions of the problem and we prove an explicit and general decay rate result, using the multiplier method and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term.

Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian

Abstract: In this paper, we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following hyperbolic problem involving the fractional Laplacian $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} +[u]^{2 (\theta -1)}_{s}(-\Delta )^su=|u|^{p-1}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\quad u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N {\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array}\right. } \end{aligned}$$ where $$[u]_{s}$$ is the Gagliardo seminorm of u, $$s\in (0,1)$$ , $$\theta \in [1, 2_s^*/2)$$ , with $$2_s^*=2N/(N-2s)$$ , $$p\in (2\theta -1, 2_s^*-1]$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary $$\partial \Omega $$ , $$(-\Delta )^s$$ is the fractional Laplacian. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blowup of solutions for the above problem by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above problem with the critical initial conditions. The significant feature and difficulty of the above problem are that the coefficient of $$(-\Delta )^s$$ can vanish at zero.

Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth

Abstract: We consider the Neumann and Cauchy problems for positivity preserving reaction–diffusion systems of m equations enjoying the mass and entropy dissipation properties. We show global classical existence in any space dimension, under the assumption that the nonlinearities have at most quadratic growth. This extends previously known results which, in dimensions $$n\ge 3$$ , required mass conservation and were restricted to the Cauchy problem. Our proof is also simpler.

Maximal function characterizations for Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type

Abstract: Let X be a metric measure space with a doubling measure and L be a nonnegative self-adjoint operator acting on $$L^2(X)$$ . Assume that L generates an analytic semigroup $$e^{-tL}$$ whose kernels $$p_t(x,y)$$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables x and y. In this article, we continue a study in Song and Yan (Adv Math 287:463–484, 2016) to give an atomic decomposition for the Hardy spaces $$ H^p_{L,\mathrm{max}}(X)$$ in terms of the nontangential maximal function associated with the heat semigroup of L, and hence, we establish characterizations of Hardy spaces associated with an operator L, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of $$ H^p_{L, \mathrm{max}}(X)$$ in terms of the radial maximal function.

The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates

Abstract: We consider the stochastic NLS with nonlinear Stratonovich noise for initial values in $${L^2({\mathbb {R}^d})}$$ and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic and stochastic Strichartz estimates. In the subcritical case we prove that the solution is global, if we impose an additional assumption on the nonlinear noise.

Extension technique for complete Bernstein functions of the Laplace operator

Abstract: We discuss the representation of certain functions of the Laplace operator $$\Delta $$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $$(-\Delta )^{1/2}$$ , the square root of the d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $$(d + 1)$$ -dimensional Laplace operator $$\Delta _{t,x}$$ in $$(0, \infty ) \times \mathbf {R}^d$$ . Caffarelli and Silvestre extended this to fractional powers $$(-\Delta )^{\alpha /2}$$ , which correspond to operators $$ abla _{t,x} (t^{1 - \alpha } abla _{t,x})$$ . We provide an analogous result for all complete Bernstein functions of $$-\Delta $$ using Krein’s spectral theory of strings. Two sample applications are provided: a Courant–Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrodinger operators $$\psi (-\Delta ) + V(x)$$ , as well as an upper bound for the eigenvalues of these operators. Here $$\psi $$ is a complete Bernstein function and V is a confining potential.

Well-posedness and stability results for nonlinear abstract evolution equations with time delays

Abstract: We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the $$C_0$$ -semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a general exponential decay estimate for small time delays if the nonlinear term is globally Lipschitz and an exponential decay estimate for solutions starting from small data when the nonlinearity is only locally Lipschitz and the linear part is a negative selfadjoint operator. In the latter case we do not need any restriction on the size of the time delays. In both cases, concrete examples are presented that illustrate our abstract results.

A note on the uniqueness of weak solutions to a class of cross-diffusion systems

Abstract: The uniqueness of bounded weak solutions to strongly coupled parabolic equations in a bounded domain with no-flux boundary conditions is shown. The equations include cross-diffusion and drift terms and are coupled self-consistently to the Poisson equation. The model class contains special cases of the Maxwell–Stefan equations for gas mixtures, generalized Shigesada–Kawasaki–Teramoto equations for population dynamics, and volume-filling models for ion transport. The uniqueness proof is based on a combination of the $$H^{-1}$$ technique and the entropy method of Gajewski.

Remark on the global regularity of 2D MHD equations with almost Laplacian magnetic diffusion

Abstract: Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regularity issue on the 2D MHD equations with standard Laplacian magnetic diffusion.

Singular sensitivity in a Keller–Segel-fluid system

Abstract: In bounded smooth domains $$\Omega \subset \mathbb {R}^N$$ , $$N\in \{2,3\}$$ , considering the chemotaxis–fluid system $$\begin{aligned} n_t + u\cdot abla n&= \Delta n - \chi abla \cdot \left( \frac{n}{c} abla c\right) \\ c_t + u\cdot abla c&= \Delta c - c + n\\ u_t + \kappa (u\cdot abla ) u&= \Delta u + abla P + n abla \phi \end{aligned}$$ with singular sensitivity, we prove global existence of classical solutions for given $$\phi \in C^2(\overline{\Omega })$$ , for $$\kappa =0$$ (Stokes-fluid) if $$N=3$$ and $$\kappa \in \{0,1\}$$ (Stokes- or Navier–Stokes-fluid) if $$N=2$$ and under the condition that $$\begin{aligned} 0<\chi <\sqrt{\frac{2}{N}}. \end{aligned}$$

Approaching nonsmooth nonconvex minimization through second-order proximal-gradient dynamical systems

Abstract: We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.

Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms

Abstract: We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.

$$\mathcal {R}$$ R -sectoriality of higher-order elliptic systems on general bounded domains

Abstract: On bounded domains $$\varOmega \subset {\mathbb {R}}^d , d \ge 2$$ , reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order 2m in divergence form with complex $$\mathrm {L}^{\infty }$$ -coefficients complemented with homogeneous mixed Dirichlet/Neumann boundary conditions. We prove that the $$\mathrm {L}^p$$ -realization of the corresponding operator A is $$\mathcal {R}$$ -sectorial of angle $$\omega \in [0 , \frac{\pi }{2})$$ , where in the case $$2m \ge d$$ , $$p \in (1,\infty )$$ , and where $$p \in (\frac{2d}{d + 2 m} - \varepsilon , \frac{2d}{d - 2 m} + \varepsilon )$$ for some $$\varepsilon > 0$$ in the case $$2m < d$$ . To perform this proof, we generalize the $$\mathrm {L}^p$$ -extrapolation theorem of Shen to the Banach space-valued setting and to arbitrary Lebesgue-measurable underlying sets.

Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms

Abstract: This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain Ω of R^N with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain ω.

On the stability in phase-lag heat conduction with two temperatures

Abstract: We investigate the well-posedness and the stability of the solutions for several Taylor approximations of the phase-lag two-temperature equations. We give conditions on the parameters which guarantee the existence and uniqueness of solutions as well as the stability and the instability of the solutions for each approximation.

Hardy inequalities, Rellich inequalities and local Dirichlet forms

Abstract: First, the Hardy and Rellich inequalities are defined for the sub-Markovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition, the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain $$\Omega \subseteq \mathbf{R}^d$$ . The weighting near the boundary $$\partial \Omega $$ can be different from the weighting at infinity. Finally, these results are applied to weighted second-order operators on $$\mathbf{R}^d\backslash \{0\}$$ and to a general class of operators of Grushin type.

Existence and comparison results for an elliptic equation involving the 1-Laplacian and $$L^1$$-data

Abstract: This paper is devoted to analyze the Dirichlet problem for a nonlinear elliptic equation involving the 1-Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the inverse mean curvature flow. We study this problem in an open bounded set with Lipschitz boundary. We prove an existence result and a comparison principle for nonnegative $$L^1$$ -data. Moreover, we search the summability that the solution reaches when more regular $$L^p$$ -data, with $$1

Strichartz estimates for harmonic potential with time-decaying coefficient

Abstract: In this paper, we prove the Strichartz estimates for the Schrodinger equations with a harmonic potential with a time-decaying coefficient.

Weighted Energy-Dissipation approach to doubly nonlinear problems on the half line

Abstract: We discuss a variational approach to abstract doubly nonlinear evolution systems defined on the time half line \(t>0\). This relies on the minimization of weighted energy-dissipation (WED) functionals, namely a family of \(\varepsilon \)-dependent functionals defined over entire trajectories. We prove WED functionals admit minimizers and that the corresponding Euler–Lagrange system, which is indeed an elliptic-in-time regularization of the original problem, is strongly solvable. Such WED minimizers converge, up to subsequences, to a solution of the doubly nonlinear system as \(\varepsilon \rightarrow 0\). The analysis relies on a specific estimate on WED minimizers, which is specifically tailored to the unbounded time interval case. In particular, previous results on the bounded time interval are extended and generalized. Applications of the theory to classes of nonlinear PDEs are also presented.

Decay of $$C_0$$-semigroups and local decay of waves on even (and odd) dimensional exterior domains

Abstract: We prove decay rates for a vector-valued function f of a nonnegative real variable with bounded weak derivative, under rather general conditions on the Laplace transform $$\hat{f}$$ . This generalizes results of Batty and Duyckaerts (J Evol Equ 8(4):765–780, 2008) and other authors in later publications. Besides the possibility of $$\hat{f}$$ having a singularity of logarithmic type at zero, one novelty in our paper is that we assume $$\hat{f}$$ to extend to a domain to the left of the imaginary axis, depending on a nondecreasing function M and satisfying a growth assumption with respect to a different nondecreasing function K. The decay rate is expressed in terms of M and K. We prove that the obtained decay rates are essentially optimal for a very large class of functions M and K. Finally, we explain in detail how our main result improves known decay rates for the local energy of waves on exterior domains.

Bounded $H_\infty$-calculus for cone differential operators

Abstract: We prove that parameter-elliptic extensions of cone differential operators have a bounded $$H_\infty $$ -calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.

Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law

Abstract: We study the Cauchy problem for a semilinear stochastic Maxwell equation with Kerr-type nonlinearity and a retarded material law. We show existence and uniqueness of strong solutions using a refined Faedo–Galerkin method and spectral multiplier theorems for the Hodge–Laplacian. We also make use of a rescaling transformation that reduces the problem to an equation with additive noise to get an appropriate a priori estimate for the solution.

Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

Abstract: In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.

Non-uniform dependence of the data-to-solution map for the Hunter–Saxton equation in Besov spaces

Abstract: The Cauchy problem for the Hunter–Saxton equation is known to be locally well posed in Besov spaces $$B^s_{2,r} $$ on the circle. We prove that the data-to-solution map is not uniformly continuous from any bounded subset of $$B^s_{2,r} $$ to $$C([0, T]; B^s_{2,r} )$$ . We also show that the solution map is Holder continuous with respect to a weaker topology.

Local null controllability of one-phase Stefan problems in 2D star-shaped domains

Abstract: In this work, we are concerned with the null controllability of one-phase Stefan problems in star-shaped domains in $$\mathbb {R}^2$$ . We prove that, for fixed $$T>0$$ and sufficient small initial data, there exist controls that drive the state to zero at time $$t=T$$ . Our approach relies on a null controllability result for parabolic systems in non-cylindrical domains, a complete analysis of the regularity of the controlled solution near the free boundary and a fixed-point argument.

Asymptotic behavior of solutions to a class of diffusive predator–prey systems

Abstract: We study the large time behavior of a class of diffusive predator–prey systems posed on the whole Euclidean space. By studying a family of similar problems with all possible spatial translations, we first prove the asymptotic persistence of the prey for the spatially heterogeneous case under certain assumptions on the coefficients. Then, applying this persistence theorem, we prove the convergence of the solution to the unique positive equilibrium for the spatially homogeneous case, under certain restrictions on the space dimension and the predation coefficient.

Sharp growth rates for semigroups using resolvent bounds

Abstract: We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an $$L^{p}$$ -space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article, we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.

Well-posedness of the vector advection equations by stochastic perturbation

Abstract: A linear stochastic vector advection equation is considered. The equation may model a passive magnetic field in a random fluid. The driving velocity field is a integrable to a certain power, and the noise is infinite dimensional. We prove that, thanks to the noise, the equation is well posed in a suitable sense, opposite to what may happen without noise.