Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a pre-publication pdf cop

Existence solutions for a weighted equation of p-biharmonic type in the unit ball of RN\mathbb{R}^{N} with critical exponential growth

We study a weighted $\frac{N}{2}$ biharmonic equation involving a positive continuous potential in $\overline{B}$. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in the unit ball of $\mathbb{R}^{N}$. It is proved that there is a nontrivial weak solution to this problem by the mountain Pass Theorem. We avoid the loss of compactness by proving a concentration compactness result and by a suitable asymptotic condition.Comment: arXiv admin note: substantial text overlap with arXiv:2201.10433, arXiv:2201.09858. substantial text overlap with arXiv:2311.1678

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

A Logarithmic Weighted Adams-type inequality in the whole of RN\mathbb{R}^{N} with an application

In this paper, we will establish a logarithmic weighted Adams inequality in a logarithmic weighted second order Sobolev space in the whole set of $\mathbb{R}^{N}$. Using this result, we delve into the analysis of a weighted fourth-order equation in $\mathbb{R}^{N}$. We assume that the non-linearity of the equation exhibits either critical or subcritical exponential growth, consistent with the Adams-type inequalities previously established. By applying the Mountain Pass Theorem, we demonstrate the existence of a weak solution to this problem. The primary challenge lies in the lack of compactness in the energy caused by the critical exponential growth of the non-linear term $f$.Comment: arXiv admin note: text overlap with arXiv:2201.1043