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Given $\mu>0$ we look for solutions $ \lambda\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -\Delta v_i+ \lambda v_i+V_i(x)v_i = \sum_{\substack{j=1}}^k\beta_{ij} v_iv_j^2 &\text{ in }…

We consider the initial value problem for the thermal-diffusive combustion systems of the form: $u_{1,t}= Delta_{x}u_1 - u_1 u_2^m$, $u_{2,t}= d Delta_{x} u_2 + u_1 u_2^m$, $x in R^{n}$, $n geq 1$, $m geq 1$, $d > 1$, with bounded uniformly…

Formulas for the solutions of initial value problems for ordinary differential equations with singular $\delta^{(n)}$-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of…

A theorem on the solutions of the problem $U'(w)=\gamma F(U(w),w),\ U(w_1)=u_2,\ U(w_2)=u_2$ is applied for finding the functional solutions of the system of partial differential equations \begin{equation} \nabla\cdot(a(u,w)\nabla u)=0,\…

We consider a parabolic-elliptic chemotaxis system generalizing \[ \begin{cases}\begin{split} & u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u(u+1)^{\sigma-1}\nabla v)\\ & 0 = \Delta v - v + u \end{split}\end{cases} \] in bounded…

This work deals with the consumption chemotaxis problem \begin{equation*} \begin{cases*} u_t = \Delta u - \chi \nabla \cdot u\nabla v + \lambda u - \mu u^2 - c \lvert \nabla u \rvert^\gamma, & \text{in $\Omega\times(0,\tmax)$}, v_t = \Delta…

In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any $s\ge 1$), $$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v \quad…

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schr\"{o}dinger type problems with supercritical exponential growth $$ -\Delta u +\lambda u+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad…

This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$$\partial_tu=\Delta u-\chi\nabla\cdot(u\nabla…

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$ u_t - u |\nabla| u + |\nabla|(u^2) = 0. $$ We construct global classical solutions starting from smooth…

For the gang territoriality model \begin{align*} \begin{cases} u_t = D_u \Delta u + \chi_u \nabla \cdot (u \nabla w), \\ v_t = D_v \Delta v + \chi_v \nabla \cdot (v \nabla z), \\ w_t = -w + \frac{v}{1+v}, \\ z_t = -z + \frac{u}{1+u},…

The paper is concerned with a scalar balance law, where the source term depends on a control function $\alpha(t)$. Given a control $\alpha\in \mathbf{L}^\infty\bigl([0,T]\bigr)$, it is proved that, for generic initial data $\bar u \in…

In a ball $\Omega\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[ \left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - uv, \end{array} \right. \] is…

We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array} [c]{lll}…

This work considers the Keller-Segel consumption system \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta (u\phi(v))+au-bu^\gamma,\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0 \end{array} \right.…

Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic…

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\…

We study the chemotaxis-fluid system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(\frac{n}{c}\nabla c),\ &x\in\Omega,& t>0, c_{t}&+&u\cdot\!\nabla…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

This paper investigates the following chemotaxis system featuring weak degradation and nonlinear motility functions \begin{equation}\label{Model1} \begin{cases} u_{t} = (\gamma(v)u)_{xx} + r - \mu u, & x \in [0,L],\ t > 0, v_{t} = v_{xx} -…