In this paper we give a detailed study on the attractors for the parabolic equations in bounded domains involving $p$-Laplacian as the principal term. Not only the existence of attractors but also their new properties are presented, which cannot be observed for the non-degenerate parabolic equations. In particular, we construct infinite-dimensional attractors whose $\varepsilon$-Kolmogorov entropy behave as the polynomial of $1/\varepsilon$ as $\varepsilon$ tends to zero.

In this paper we provide examples of blowing-up solutions to parabolic problems in a half space, ${{\mathbb{R}}}^N_+ \times {{\mathbb{R}}}^M = \{x_N >0 \} \times {{\mathbb{R}}}^M$, with nontrivial blow-up sets of dimension strictly smaller than the space dimension. To this end we prove existence of a nontrivial compactly supported solution to $\nabla (|\nabla \varphi|^{p-2} \nabla \varphi) = \varphi $ in the half space ${{\mathbb{R}}}^N_+ =\{x_N >0\}$ with the nonlinear boundary condition $-|\nabla \varphi|^{p-2} \frac{\partial \varphi}{\partial x_N} = \varphi^{p-1}$ on $\partial {{\mathbb{R}}}^N_+ =\{ x_N =0\}$.

The $I$-method in its first version as developed by Colliander et~al. in [2] is applied to prove that the Cauchy-problem for the generalized Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space $H^s(\mathbb R \rightarrow \mathbb R)$, provided $s>-\frac{1}{42}$.

The effect of the domain topology on the multiplicity of solutions for a semilinear elliptic system with critical Sobolev exponent is discussed. We show that if the coupling term is sufficiently small, then there exist at least cat $\Omega$ positive solutions, where cat denotes the Ljusternik-Schnirelman category.

In this paper, we consider elliptic and parabolic equations with unbounded coefficients in smooth exterior domains $\Omega\subset {\mathbb R}^N$, subject to Dirichlet or Neumann boundary conditions. Under suitable assumptions on the growth of the coefficients, the solution of the parabolic problem is governed by a semigroup $\{T(t)\}$ on $L^p(\Omega)$ for $1 < p < \infty$ and on $C_b(\overline\Omega)$. Furthermore, uniform- and $L^p$-estimates for higher-order spatial derivatives of $\{T(t)\}$ are obtained. They imply optimal Schauder estimates for the solution of the corresponding elliptic and parabolic problems.

In this paper we consider some dissipative versions of the modified Korteweg--de~Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0 <\alpha\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces $H^s({ \mathbb R})$ for $s>1/4-\alpha/4$ on the basis of the $[k;\,Z]-$multiplier norm estimate obtained by Tao in [11] for KdV equation.