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Abstract: We consider the Cauchy problem for the intermediate long-wave equation $$\begin{aligned} u_{t}-\partial _{x}u^{2}+\frac{1}{\vartheta }u_{x}+VP\int _{\mathbb {R}}\frac{1}{2\vartheta }\coth \left( \frac{\pi \left( y-x\right) }{2\vartheta }\right) u_{yy}\left( t,y\right) \mathrm{d}y=0, \end{aligned}$$ where \(\vartheta >0\) . Our purpose in this paper is to prove the large time asymptotic behavior of solutions under the nonzero mass condition \(\int u_{0}\left( x\right) \mathrm{d}x\ne 0\) . PubDate: 2019-03-22 DOI: 10.1007/s00028-019-00498-5

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Abstract: For any bounded smooth domain \(\Omega \subset \mathbb {R}^n\) , we establish the global existence of weak solutions \(u\in L^2(0,T;H^s_{\Omega ,0})\) with \(u_t\in L^2(0,T;H^{-s}_{\Omega })\) to the initial boundary value problem of the nonlocal energy-weighted fractional reaction–diffusion equations for any initial data \(u_0\in H^s_{\Omega ,0}\) . PubDate: 2019-03-08 DOI: 10.1007/s00028-019-00494-9

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Abstract: We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming that the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times. PubDate: 2019-03-07 DOI: 10.1007/s00028-019-00493-w

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Abstract: We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space \({\mathbb {R}}^N\) . We show that given a weighted \(L^p\) -space \(L_w^p({\mathbb {R}}^N)\) with \(1 \le p < \infty \) and a fast-growing weight w, there are Schauder bases \((e_n)_{n=1}^\infty \) in \(L_ w^p({\mathbb {R}}^N)\) with the following property: given a positive integer m, there exists \(n_m > 0\) such that, if the initial data f belong to the closed linear space of \(e_n\) with \(n \ge n_m\) , then the decay rate of the solution of the heat equation is at least \(t^{-m}\) . Such a basis can be constructed as a perturbation of any given Schauder basis. The proof is based on a construction of a basis of \(L_w^p( {\mathbb {R}}^N)\) , which annihilates an infinite sequence of bounded functionals. PubDate: 2019-03-02 DOI: 10.1007/s00028-019-00492-x

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Abstract: For a class of Laplace exponents, we consider the transition density of the subordinator and the heat kernel of the corresponding subordinate Brownian motion. We derive explicit approximate expressions for these objects in the form of asymptotic expansions: via the saddle point method for the subordinator’s transition density and via the Mellin transform for the subordinate heat kernel. The latter builds on ideas from index theory using zeta functions. In either case, we highlight the role played by the analyticity of the Laplace exponent for the qualitative properties of the asymptotics. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0468-9

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Abstract: Let \(\Omega \) be a bounded open subset with \(C^{1+\kappa }\) -boundary for some \(\kappa > 0\) . Consider the Dirichlet-to-Neumann operator associated with the elliptic operator \(- \sum \partial _l ( c_{kl} \, \partial _k ) + V\) , where the \(c_{kl} = c_{lk}\) are Hölder continuous and \(V \in L_\infty (\Omega )\) are real valued. We prove that the Dirichlet-to-Neumann operator generates a \(C_0\) -semigroup on the space \(C(\partial \Omega )\) which is in addition holomorphic with angle \(\frac{\pi }{2}\) . We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence, we obtain an optimal holomorphic functional calculus and maximal regularity on \(L_p(\Gamma )\) for all \(p \in (1,\infty )\) . PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0467-x

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Abstract: In the slow diffusion case, unbounded supersolutions of the porous medium equation are of two totally different types, depending on whether the pressure is locally integrable or not. This criterion and its consequences are discussed. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0474-y

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Abstract: Our aim is the study of well-posedness for the stochastic evolution equation $$\begin{aligned} \mathrm{d}u -\mathrm {div}\,( \nabla u ^{p-2}\nabla u +F(u)) \,\mathrm{d} t = H(u) \,\mathrm{d}W, \end{aligned}$$ for \(T>0\) , on a bounded Lipschitz domain \(D \subset \mathbb {R}^d\) with homogeneous Dirichlet boundary conditions, initial values \(u_0\in L^2(D)\) , \(p>2\) and \(F:\mathbb {R}\rightarrow \mathbb {R}^d\) Lipschitz continuous. W(t) is a cylindrical Wiener process in \(L^2(D)\) with respect to a filtration \((\mathcal {F}_t)\) satisfying the usual assumptions on a complete, countably generated probability space \((\varOmega ,\mathcal {F},\mathbb {P})\) . We consider the case of multiplicative noise satisfying appropriate regularity conditions. By a semi-implicit time discretization, we obtain approximate solutions. Using the theorems of Skorokhod and Prokhorov, we are able to pass to the limit and show existence of martingale solutions. We establish pathwise \(L^1\) -contraction and uniqueness and obtain existence and uniqueness of strong solutions. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0472-0

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Abstract: We consider the weighted parabolic problem of the type $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} u_t-\mathrm{div}(\omega _2(x) \nabla u ^{p-2} \nabla u )= \lambda \omega _1(x) u ^{p-2}u,&{} x\in \Omega ,\\ u(x,0)=f(x),&{} x\in \Omega ,\\ u(x,t)=0,&{} x\in \partial \Omega ,\ t>0,\\ \end{array}\right. \end{aligned} \end{aligned}$$ for quite a general class of possibly unbounded weights \( \omega _1,\omega _2\) satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that \(\lambda >0\) is smaller than the optimal constant in the inequality. The domain is assumed to be bounded or quasibounded. The obtained solution is proven to belong to $$\begin{aligned} L^p({{\mathbb {R}}}_+; W_{(\omega _1,\omega _2),0}^{1,p}(\Omega ))\cap L^\infty ({{\mathbb {R}}}_+; L^2(\Omega )). \end{aligned}$$ PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0465-z

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Abstract: We consider the simplest parabolic–elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the blowup of radially symmetric solutions in terms of suitable Morrey spaces norms are derived. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0469-8

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Abstract: We consider the well-posedness of a class of hyperbolic partial differential equations on a one-dimensional spatial domain. This class includes in particular infinite networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. We consider these equations on a finite interval as well as on a semi-axis. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0470-2

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Abstract: We study the following time-fractional nonlinear superdiffusion equation $$\begin{aligned} \left\{ \begin{array}{l} {}_0^CD_t^\alpha u-\triangle u= u ^{p}, \ \ x\in \mathbb {R}^N,\ \ t>0, \\ u(0,x)=u_0(x),\ \ u_t(0,x)=u_1(x),\ \ x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$ where \(1<\alpha <2\) , \(p>1\) , \(u_0,u_1\in L^q(\mathbb {R}^N)\) ( \(q>1\) ) and \({_0^CD_t^\alpha u}\) denotes the Caputo fractional derivative of order \(\alpha .\) The critical exponents of this problem are determined when \(u_1\equiv 0\) and \(u_1\not \equiv 0, \) respectively. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0475-x

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Abstract: We consider the Navier–Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space–time-dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier–Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0473-z

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Abstract: We study functions of bounded variation with values in a Banach or in a metric space. In finite dimensions, there are three well-known topologies; we argue that in infinite dimensions there is a natural fourth topology. We provide some insight into the structure of these four topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin–Lions theorem. After this we study the Borel \(\sigma \) -algebras induced by these topologies, and we provide some results about probability measures on the space of functions of bounded variation, which can be used to study stochastic processes of bounded variation. PubDate: 2019-03-01 DOI: 10.1007/s00028-018-0471-1

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Abstract: Let X be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure \(\gamma \) , and let \(\lambda _1\) be the maximum eigenvalue of the covariance operator associated with \(\gamma \) . The associated Cameron–Martin space is denoted by H. For a sufficiently regular convex function \(U:X\rightarrow {{\mathbb {R}}}\) and a convex set \(\Omega \subseteq X\) , we set \(\nu :=\hbox {e}^{-U}\gamma \) and we consider the semigroup \((T_\Omega (t))_{t\ge 0}\) generated by the self-adjoint operator defined via the quadratic form $$\begin{aligned} (\varphi ,\psi )\mapsto \int _\Omega {\left\langle D_H\varphi ,D_H\psi \right\rangle }_H \hbox {d}\nu , \end{aligned}$$ where \(\varphi ,\psi \) belong to \(D^{1,2}(\Omega ,\nu )\) , the Sobolev space defined as the domain of the closure in \(L^2(\Omega ,\nu )\) of \(D_H\) , the gradient operator along the directions of H. A suitable approximation procedure allows us to prove some pointwise gradient estimates for \((T_{\Omega }(t))_{t\ge 0}\) . In particular, we show that $$\begin{aligned} D_H T_{\Omega }(t)f _H^p\le \hbox {e}^{- p \lambda _1^{-1} t}(T_{\Omega }(t) D_H f ^p_H), \quad \, t>0,\ \nu \text {-a.e. in }{\Omega }, \end{aligned}$$ for any \(p\in [1,+\infty )\) and \(f\in D^{1,p}({\Omega },\nu )\) . We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in \({\Omega }\) for the measure \(\nu \) and some improving summability properties for \((T_\Omega (t))_{t\ge 0}\) . In addition, we prove that if f belongs to \(L^p(\Omega ,\nu )\) for some \(p\in (1,\infty )\) , then $$\begin{aligned} D_H T_\Omega (t)f ^p_H \le K_p t^{-\frac{p}{2}} T_\Omega (t) f ^p,\quad \, t>0,\ \nu \text {-a.e. in }\Omega , \end{aligned}$$ where \(K_p\) is a positive constant depending only on p. Finally, we investigate on the asymptotic behaviour of the semigroup \((T_{\Omega }(t))_{t\ge 0}\) as t goes to infinity. PubDate: 2019-02-22 DOI: 10.1007/s00028-019-00491-y

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Abstract: This paper is concerned with estimates of the gradient of the solutions to the Stokes IBVP both in a bounded and in an exterior domain. More precisely, we look for estimates of the kind \( \nabla v(t) _q \le g(t) \nabla v_0 _p,\;q\ge p>1,\) for all \(t>0\) , where function g is independent of v. PubDate: 2019-02-12 DOI: 10.1007/s00028-019-00490-z

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Abstract: In this paper, we consider the following Kirchhoff-type wave problems, with nonlinear damping and source terms involving the fractional Laplacian, $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{tt} +[u]^{2\gamma -2}_{s}(-\Delta )^su+ u_t ^{a-2}u_t+u= u ^{b-2}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\ \ \ \ u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N{\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array} \right. \end{aligned}$$ where \((-\Delta )^s\) is the fractional Laplacian, \([u]_{s}\) is the Gagliardo semi-norm of u, \(s\in (0,1)\) , \(2<a<2\gamma<b<2_s^*=2N/(N-2s)\) , \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary \(\partial \Omega \) . Under some natural assumptions, we obtain the global existence, vacuum isolating, asymptotic behavior and blowup of solutions for the problem above by combining the Galerkin method with potential wells theory. The significant feature and difficulty of the problem are that the coefficient of \((-\Delta )^s\) can vanish at zero. PubDate: 2019-02-12 DOI: 10.1007/s00028-019-00489-6

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Abstract: In this paper, we consider an abstract viscoelastic equation with minimal conditions on the \(L^{1}(0,\infty )\) relaxation function g namely \( g^{\prime }(t)\le -\xi (t)H(g(t))\) , where H is an increasing and convex function near the origin and \(\xi \) is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when \(H(s)=s^{p}\) and p covers the full admissible range [1, 2). We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature. PubDate: 2019-02-08 DOI: 10.1007/s00028-019-00488-7

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Abstract: We propose a probabilistic construction for the solution of a general class of fractional high-order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order \(\alpha \in (0,1)\) . PubDate: 2019-02-02 DOI: 10.1007/s00028-019-00485-w

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Abstract: This work investigates the formation of singularities under the steepest descent \(L^2\) -gradient flow of \({{\,\mathrm{{ W}}\,}}_{\lambda _1, \lambda _2}\) with zero spontaneous curvature, i.e., the sum of the Willmore energy, \(\lambda _1\) times the area, and \(\lambda _2\) times the signed volume of an immersed closed surface without boundary in \(\mathbb {R}^3\) . We show that in the case that \(\lambda _1>1\) and \(\lambda _2=0\) , any immersion develops singularities in finite time under this flow. If \(\lambda _1 >0\) and \(\lambda _2 > 0\) , embedded closed surfaces with energy less than $$\begin{aligned} 8\pi +\min \left\{ \left( 16 \pi \lambda _1^3\right) \bigg /\left( 3\lambda _2^2\right) , 8\pi \right\} \end{aligned}$$ and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than \(8 \pi \) , the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that \(\lambda _2\) is negative. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For \(\lambda _1 >0\) and \(\lambda _2 \ge 0\) , they showed that embedded closed spheres with positive volume and energy close to \(4\pi \) , i.e., close to the Willmore energy of a round sphere, converge to round points in finite time. PubDate: 2019-02-01 DOI: 10.1007/s00028-019-00483-y