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In this paper, we prove an interpolation inequality on Riemann–Liouville fractional integrals and then use it to study the strong stability and semi-uniform stability of fractional resolvent families of order 0 < α < 2 . Let A denote the generator of a bounded fractional resolvent family. We show that if σ (A) ∩ (i R) α is countable and σ r (A) ∩ (i R) α = ∅ , then the bounded fractional resolvent family is strongly stable. And the semi-uniform stability of the fractional resolvent family is equivalent to σ (A) ∩ (i R) α = ∅ . Moreover, the relation between decay rates of semi-uniform stability and growth of the resolvent of A along (i R) α is given. [ABSTRACT FROM AUTHOR]

Let m∈Nm\in {\mathbb{N}} and be a generalized Orlicz function. We obtained some interpolation inequalities for derivatives in generalized Orlicz-Sobolev spaces Wm,φ(Rn){W}^{m,\varphi }\left({{\mathbb{R}}}^{n}). As applications, we established a compact Sobolev embedding on domain and a Landau-Kolmogorov-type inequality in generalized Orlicz spaces. And we introduced the Sobolev φ\varphi -capacity and studied some of its properties.

We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p norms in combination with an L p estimate with low exponent p > 1 . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$ and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$. We also prove an existence result of standing waves and prove their orbital stability. [ABSTRACT FROM AUTHOR]

The classical Gagliardo-Nirenberg inequality was established in Rn. An extension to a bounded domain was given by Gagliardo in 1959. In this note, we present a simple proof of this result and prove a new Gagliardo-Nirenberg inequality in a bounded Lipschitz domain. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the nonlinear Bürgers equation and prove a new interpolation inequality. The method of energy inequalities involving the new interpolation inequality is used to study the solvability of the problem in question. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the energy decay for solutions of the weakly coupled dissipative Schrödinger system. Among the m-coupled equations, only one equation is directly damped. Under some assumptions about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically with mixed boundary conditions, including the coupling of the Schrödinger system subject to Dirichlet and Robin type boundary conditions, respectively. The proof is based on some frequency estimates with an exponential loss on the resolvent operators, which will be solved by establishing an interpolation inequality for a suitable weakly coupled elliptic system. [ABSTRACT FROM AUTHOR]

Motivated by the studies of the hydrodynamics of the tethered bacteria Thiovulum majus in a liquid environment, we consider the following chemotaxis system under homogeneous Neumann boundary conditions in a bounded convex domain with smooth boundary. For any given fluid , it is proved that if , the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if , such solution still exists under the additional condition that. [ABSTRACT FROM AUTHOR]

We prove that for some initial data, solutions of the Cauchy problem for the nonlinear Schrödinger equation in the supercritical case are destroyed after a finite time, the exact value of which can be estimated from above. Lower bounds are obtained for the rate of destruction of the solution in some norms. A set of initial data is identified for which the solution of the Cauchy problem for the nonlinear Schrödinger equation in the supercritical case exists globally. [ABSTRACT FROM AUTHOR]

In this paper, we consider a damped wave equation with mixed boundary conditions in a bounded domain. On one portion of the boundary, we have kinetic boundary condition: ∂ ν y + m (x) y tt - Δ T y = 0 with density function m(x), and on the other portion, we have homogeneous Neumann boundary condition: ∂ ν y = 0 . Based on the growth of the resolvent operator on the imaginary axis, solutions of the wave equations under consideration are proved to decay logarithmically. The proof of the resolvent estimation relies on the interpolation inequalities for an elliptic equation with Steklov type boundary conditions. [ABSTRACT FROM AUTHOR]

A carefully written Nirenberg's proof of the famous Gagliardo-Nirenberg interpolation inequality for intermediate derivatives in Rn seems, surprisingly, to be missing in literature. In our paper, we shall first introduce this fundamental result and provide information about its historical background. Afterwards, we present a complete, student-friendly proof. In our proof, we use the architecture of Nirenberg's argument, the explanation is, however, much more detailed, also containing some differences. The reader can find a short comparison of differences and similarities in the final chapter. [ABSTRACT FROM AUTHOR]

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems related to integral estimates and regularity of solutions to elliptic and/or parabolic equations. The main purpose of our work is to provide an important observation concerning the -boundedness property in the context of interpolation inequalities between Sobolev and Morrey spaces, which may be useful for those working in this domain. We also construct a nontrivial counterexample, which shows that the range of admissible values of is optimal in a certain sense. Our proofs rely on integral representations and on the theory of maximal and sharp maximal functions. [ABSTRACT FROM AUTHOR]

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms. [ABSTRACT FROM AUTHOR]

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called "clamped" boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously. We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp(Cμ1/4) where μ is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application, we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator, in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument. [ABSTRACT FROM AUTHOR]

It is proved that, for some initial data, the solutions of the Cauchy problem for the cubic Schrödinger evolution equation blow up in finite time whose exact value is estimated from above. In addition, lower bounds for the blow-up rate of the solution in certain norms are obtained. [ABSTRACT FROM AUTHOR]

We mainly investigate the global boundedness of the solution to the following system, ut=u-·(uv)in O×R+, vt=v-v+w, in O×R+, wt=w-w+u, in O×R+, under homogeneous Neumann boundary conditions with nonnegative smooth initial data in a smooth bounded domain ORn with critical space dimension n=4. This problem has been considered by K. Fujie and T. Senba in [5]. They proved that for the symmetric case the condition Ou0<(8p)²/yields global boundedness, where u0 is the instal data for u. In this paper, inspired by some new techniques established in [3], we give a new criterion for global boundedness of the solution. As a byproduct, we obtain a simplified proof for one of the main results in [5]. [ABSTRACT FROM AUTHOR]

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces. [ABSTRACT FROM AUTHOR]

In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.