Related papers: Regularity, asymptotic behavior and partial unique…

The processes of simultaneous coagulation and Ostwald ripening of particles in the concluding stage of phase transformation are considered. We solve the integro-differential system of Smoluchowski-type kinetic and mass balance equations…

We show uniqueness in law for the critical SPDE $$ dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, $$ where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having…

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular/smooth kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse…

In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} - \nabla \cdot a \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - a \cdot \nabla^2 + b \cdot…

We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with…

Let the set $\Omega_\varepsilon$ be obtained from the bounded domain $\Omega$ by removing a family of $\varepsilon$-periodically distributed identical balls. In $\Omega_\varepsilon$ one considers the standard Steklov spectral problem. It is…

We study a behavior of the conformal Laplacian operator $\L_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the…

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…

We introduce a one-dimensional stochastic system where particles perform independent diffusions and interact through pairwise coagulation events, which occur at a nontrivial rate upon collision. Under appropriate conditions on the diffusion…

This paper deals with the \emph{integral} version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the H\"older regularity of the data. By…

We rereview the classical Cauchy-Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy-Kovalevskaya and…

This work outlines an exact combinatorial approach to finite coagulating systems through recursive equations and use of generating function method. In the classic approach the mean-field Smoluchowski coagulation is used. However, the…

For $n\ge 3$, $0<m<\frac{n-2}{n}$, $\beta<0$ and $\alpha=\frac{2\beta}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in…

Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a…

The Smoluchowski coagulation equation (SCE) is a population balance model that describes the time evolution of cluster size distributions resulting from particle aggregation. Although it is formally a mass-conserving system, solutions may…

We study the asymptotic behaviour of solutions of a class of linear non-local measure-valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic like behaviour in the large…

We investigate a coagulation-fragmentation equation with boundary data, establishing the well-posedness of the initial value problem when the coagulation kernels are bounded at zero and showing existence of solutions for the singular…

The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness of bounded sequences. For every bounded sequence in…

We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_q^s u&=\frac{f(x)}{u^{\delta}}\text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0…

The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly…