Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation

Decay estimates of solutions to the IBq equation

‎In this paper we focus on the Cauchy problem for the generalized‎ ‎IBq equation with damped term in $n$-dimensional space‎. ‎We establish the global existence and decay estimates of solution with $L^q(1leq qleq 2)$ initial value‎, ‎provided that the initial value is suitably small‎. ‎Moreover‎, ‎we also show that the solution is asymptotic to the solution $u_L$ to the corresponding linear equa...

Sharp Estimates for Maximal Operators Associated to the Wave Equation

The wave equation, ∂ttu = ∆u, in R, considered with initial data u(x, 0) = f ∈ H(R) and u′(x, 0) = 0, has a solution which we denote by 1 2 (e √ −∆f + e−it √ −∆f). We give almost sharp conditions under which sup0<t<1 |e ±it √ −∆f | and supt∈R |e ±it √ −∆f | are bounded from H(R) to L(R).

Sharp estimates of bounded solutions to a second order forced equation with structural damping

Sobolev space estimates for solutions of the equation ∂¯u=f on polycylinders

In trying to improve Weinstock's results on approximation by holomorphic functions on certain product domains, we are led to estimates in Sobolev spaces for the ∂-operator on polycylinders for (γ, q)-forms. This generalizes our results for the same operator on poly-cylinders previously obtained, and can be applied to a number of other problems such as the Corona problem. 1. Introduction. Had we...

An Elementary Proof of Sharp Sobolev Embeddings

We present an elementary uniied and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brrzis and Wainger. Let be an open subset of R n , where n 2, let 1 p < 1 and let W 1;p (() be the Sobolev space, that is, the set of all functions in L p ((), whose distributional derivatives of the r...