In this paper, we deal with weakly coupled elliptic systems ${\mathcal A}$ with unbounded coefficients. We prove the existence and characterize all the systems of invariant measures for the semigroup $({\bf T}(t))_{t\ge 0}$ associated with ${\mathcal A}$ in $C_b({\mathbb R^d};\mathbb R^m)$. We also show some relevant properties of the extension of $({\bf T}(t))_{t\ge 0}$ to the $L^p$-spaces related to systems of invariant measures. Finally, we study the asymptotic behaviour of $({\bf T}(t))_{t\ge 0}$ as $t$ tends to $+\infty$.

In this paper, we prove the existence of infinitely many nontrivial solutions for the class of $ (p,q) $ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_{\Omega}(\Omega)$ nonnegative solutions.

This paper deals with a quasilinear parabolic equation with nonlinear gradient absorption \begin{equation*} u_t-\Delta_{p}u=u^q-\mu u^{r}|\nabla u|^\delta, \ x\in\Omega, t>0. \end{equation*} Here, $\Delta_{p} u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplace operator, $\Omega \subset \mathbb{R}^{N}$ $(N \geq 1)$ is a bounded smooth domain. By a regularization approach, we first establish the local-in-time existence of its weak solutions. Then we prove the global existence by constructing a family of bounded super-solutions which technically depend on the inradius of $\Omega$. We also obtain an upper bound and a lower bound of the blowup time. We use a comparison with suitable self-similar sub-solutions to prove the blowup and an upper bound of blowup time. Finally, we derive a lower bound of the blowup time by using the differential inequality technique.