Let $ {\Omega}\subset {\mathbb R}^N$ be a bounded domain with $C^2$ boundary. Let $u\in W^{1,N}(\Omega)$ be a weak solution of the following problem: $$(P_{\mu,{\lambda}})\hspace{1cm} \left \{ \begin{array}{cllll} \left . \begin{array}{rllll} -\text{div}(|{\nabla} u|^{N-2}{\nabla} u) +|u|^{N-2}u & =& \mu h(u) e^{u^{\alpha}}\\ u&<& 0 \end{array} \right \} \;\; \text{in} \; {\Omega}, \\ \hspace{2.2cm}|{\nabla} u|^{N-2}\frac{{\partial} u}{{\partial} \nu}\; =\;\; {\lambda} \psi u^q \;\;\text{on}\;\; \partial {\Omega}, \end{array} \right . $$ where ${\alpha}\in (0,\frac{N}{N-1}], {\lambda}, \mu >0, q\in [0,N-1)$ and $\psi$ is a positive Hölder continuous function on $\overline{ {\Omega}}$. Here, $h(u)$ is a ``suitable" perturbation of $e^{u^\alpha}$ as $u \to \infty$ (see assumptions $(\mathbf{A1})-(\mathbf{A5})$ in Section 1). In this article, we show that there exists a region $\Re \subset \{(\mu,{\lambda}): \mu,{\lambda}>0\}$ bounded by the graph of a map $ {\Lambda}$ such that $(P_{\mu,{\lambda}})$ admits at least two solutions for all $(\mu,{\lambda}) \in \Re$, at least one solution for any $(\mu,{\lambda}) \in \partial \Re$ and no solution for $(\mu,{\lambda})$ outside $\overline{\Re}$.

In this paper we provide an elementary proof of the classical result of J.L. Lions and G. Prodi on the global unique solvability of two-dimensional Navier-Stokes equations that avoids compact embedding and strong convergence. The method applies to unbounded domains without special treatment. The essential idea is to utilize the local monotonicity of the sum of the Stokes operator and the inertia term. This method was first discovered in the context of stochastic Navier-Stokes equations by J.L. Menaldi and S.S. Sritharan.

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a damped forced nonlinear Schrödinger equation in $2D$. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite-dimensional global attractor for this dynamical system.

A variational principle is introduced to provide a new formulation and resolution for several boundary-value problems. Indeed, we consider systems of the form \begin{eqnarray*} \left\{ \begin{array}{ll} \Lambda u = \nabla \Phi (u), \\ \beta_2 u= \nabla \Psi (\beta_1 u), \end{array} \right. \end{eqnarray*} where $\Phi$ and $\Psi$ are two convex functions and $\Lambda$ is a possibly unbounded self-adjoint operator modulo the boundary operator ${\mathcal B}= (\beta_1, \beta_2).$ We shall show that solutions of the above system coincide with critical points of the functional $$ I(u)= \Phi^* (\Lambda u)-\Phi(u)+\Psi^* (\beta_2 u)- \Psi(\beta_1 u), $$ where $\Phi^*$ and $\Psi^*$ are the Fenchel-Legendre dual of $\Phi$ and $\Psi$ respectively. Note that the standard Euler-Lagrange functional corresponding to the system above is of the form, $$ F(u)= \tfrac{1}{2} \langle \Lambda u, u \rangle- \Phi(u)-\Psi (\beta_1 u). $$ An immediate advantage of using the functional $I$ instead of $F$ is to obtain more regular solutions and also the flexibility to handle boundary-value problems with nonlinear boundary conditions. Applications to Hamiltonian systems and semi-linear Elliptic equations with various linear and nonlinear boundary conditions are also provided.

We prove that the Cauchy problem for a nonlinear Dirac equation with vector self-interaction (Thirring model) and for a nonlinear system of two Dirac equations coupled through a vector-vector interaction (Federbusch model) are locally well posed, in one space dimension, for initial data in Sobolev spaces of almost critical dimension; i.e., in $H^\varepsilon$, the critical space being $L^2$, and globally well posed for initial data in $H^{1/2+\varepsilon}$, for any $\varepsilon>0$. We also consider a nonlinear Dirac equation with quadratic nonlinearity which was studied earlier by S.~Machihara and N.~Bournaveas. We prove that the Cauchy problem for this equation is locally well posed for initial data in $H^{\varepsilon}$.

In this paper, we use the extrapolation method combined with a fixed-point theorem for the sum of completely continuous and contraction operators, to establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some classes of nondensely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces with infinite delay.

Stochastic non-Newtonian fluids with multiplicative noise are studied under the case of shear thinning and shear thickening and the existence of martingale solutions and stationary solutions is achieved for the first time.

We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as generalizations of semilinear reaction-diffusion equations with dynamic boundary conditions and various other phase-field models, such as the isothermal Allen-Cahn equation with dynamic boundary conditions. We thus formulate a class of initial and boundary-value problems whose global existence and uniqueness is proven by means of an appropriate Faedo-Galerkin approximation scheme developed for problems with dynamic boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor.

A non-compressible viscoelastic fluid with memory is considered and the well posedness of the associated dynamical problem is studied. The exponential decay of the energy is obtained under suitable hypotheses on the memory kernel. Furthermore, a domain of dependence inequality is established, thereby showing the hyperbolic nature of the problem.

This paper is concerned with global well posedness and the existence of attractors for the three-dimensional viscous primitive equations, describing large-scale oceanic motion under random wind stress. We prove global well posedness of the primitive equations with white-noise boundary conditions. Moreover, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.