This paper is devoted to investigating blow-up of solutions to initial boundary value problem of the Moore-Gibson-Thompson (MGT) equations on exterior domain in two dimensions. More precisely, upper bound lifespan estimates of solutions to the problem with power nonlinearity $|u|^{p}$, derivative nonlinearity $|u_{t}|^{p}$ and combined nonlinearities $|u_{t}|^{p} + |u|^{q}$ in the sub-critical and critical cases are established, respectively. The proofs are based on the new test function technique. Our main new contribution is that we improve the results in [36] in two dimensions. It is worth to mention that lifespan estimates of solutions are associated with the well-known Strauss and Glassey conjecture. Moreover, the variation trend of wave for the Cauchy problem of linear MGT equation with different initial values in two dimensions is showed by numerical simulation.

A Cauchy problem for a modification of the Swift-Hohenberg equation in $\mathbb R^{N}$ with a mildly integrable potential is considered. Existence of exponential attractors containing a finite-dimensional global attractor in $H^2(\mathbb R^{N})$ is shown under the dissipative mechanism of fourth order parabolic equations in unbounded domains.

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counterpart with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u} + (\boldsymbol{u}\cdot\nabla)\boldsymbol{u} + \alpha\boldsymbol{u} + \beta|\boldsymbol{u}|^{r-1} \boldsymbol{u} + \nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0, $$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$ and $d=3$ with $r > 3$ (for any $\beta,\mu > 0$), and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with linear multiplicative white noise. In addition, a remark on the existence of a unique weak pullback mean random attractor for the stochastic CBF equations is also provided. Finally, we establish the upper semicontinuity of the random pullback attractors, that is, the random pullback attractors converge towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the random dynamical system is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations on whole space.

We investigate zero energy critical points for a class of functionals $\Phi_\mu$ defined on a uniformly convex Banach space, and depending on a real parameter $\mu$. More precisely, we show the existence of a sequence $(\mu_n)$ such that $\Phi_{\mu_n}$ has a pair of critical points $\pm u_n$ satisfying $\Phi_{\mu_n}(\pm u_n)=0$, for every $n$. In addition, we provide some properties of $\mu_n$ and $u_n$. This result, which is proved by combining the nonlinear generalized Rayleigh quotient method [10] with the Ljusternik-Schnirelman theory, is then applied to several classes of elliptic pdes.

Consider in a bounded domain $\Omega \subset \mathbb{R}^N$, $N\ge 2$, with smooth boundary $\partial \Omega$, the following eigenvalue problem\begin{eqnarray}& ~ & -\Delta_p u-\Delta_q u=\lambda a(x) | u | ^{r-2}u\ \ \mbox{ in}~ \Omega, \nonumber \\ & ~ & \big( | \nabla u | ^{p-2}+ | \nabla u | ^{q-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) | u | ^{r-2}u~ \mbox{ on}~ \partial \Omega, \nonumber\end{eqnarray}where $1 < q < r < p < \infty,$ with $r < q(N-1)/(N-q)$ if $q < N$; $a\in L^{\infty}(\Omega),\ b\in L^{\infty}(\partial\Omega)$ are given nonnegativefunctions satisfying $\int_\Omega a~dx+\int_{\partial\Omega} b\,d\sigma > 0.$Under these assumptions, we prove that there exist two positive constants $\lambda_* < \lambda^*$ such that any $\lambda\in \{0\}\cup [\lambda^*, \infty)$ is an eigenvalue of this problem, while the set$(-\infty, 0)\cup (0, \lambda_*)$ contains no eigenvalue of the problem. This result is complementary to previous results related to the above eigenvalue problem.

In this paper, we consider the Cauchy problem forscale-invariant semilinear wave models withtime-dependent mass, dissipation and integrabletime-dependent propagation speed. The goal is tostudy the interplay between the coefficientsappearing in the mass and dissipation term andthe exponent in the speed of propagation to proveglobal (in time) existence of small dataSobolev solutions and blow-up results.

This work studies the convergence problem for a class of fractional diffusion equations in which the time-derivative order approaches $1^-$. Up to now, few works have investigated this topic. The purpose of the article consists of three main contents. The first result is related to the convergence of the Caputo derivative and the Mittag-Leffler operators when $\alpha \to 1^-$. The second is to investigate the convergence problem for a linear fractional diffusion equation on $L^p$ spaces. And last result is concerned with the convergence problem for nonlinear fractional diffusion equations. The main analysis and techniques of the paper involve the evaluation related to Riemann-Liouville integration, Caputo derivative and Sobolev embeddings. Our analysis provides a complete and detailed answer to the convergence problem as fractional order tends to $1^-$.