We study the existence and the asymptotic behavior of solutions of {1.1), when $V$ can vanish and decay to zero at infinity.

The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem \[u_t + u_{xxx} + (u^4)_x=0,\hspace{1cm}u(0)=u_0 \in H^s_x({\bf R}), \,\,\,s>-\tfrac{1}{6}\] for the generalized Korteweg-deVries equation of order three, for short gKdV-3. For real-valued data $u_0 \in L^2_x({\bf R})$ global wellposedness follows by the conservation of the $L^2$ norm. The main new tool is a bilinear estimate for solutions of the Airy-equation.

We study nonlinear pseudodifferential equations on a half-line with a nonanalytic symbol \begin{equation*} \left\{ \begin{array}{c} \partial _{t}u+\mathbb{K}u=\lambda \left\vert u\right\vert ^{\sigma }u,\text{ }x\in \mathbf{R}^{+},\text{ }t>0, \\ u\left( 0,x\right) =u_{0}\left( x\right) \text{, }x\in \mathbf{R}^{+}, \end{array} \right. \end{equation*} where $0<$ $\sigma <1,$ $\lambda \in \mathbf{R}$ and \begin{equation*} \mathbb{K}u=\frac{1}{2\pi i}\theta (x)\int_{-i\infty }^{i\infty }e^{px}K(p) \widehat{u}(t,p)dp,\qquad K(p)=\frac{p^{2}}{p^{2}-1}. \end{equation*} The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem and to find the main term of the asymptotic representation of solutions in subcritical case, when the nonlinear term of equation has the time decay rate less than that of the linear terms.

We prove boundedness of weak solutions $u$ to some linear elliptic equations in divergence form $$-div(a(x)Du(x))=\mu\,.$$ On the right-hand side we have a Radon measure $\mu$ with a suitable decay on balls $ 0\leq\mu(B(x,\rho))\leq c\rho^{s}. $

In this paper, we consider Hessian estimates of solutions of the Cauchy problem for parabolic PDEs with the Ornstein-Uhlenbeck operator. Our upper estimate on the Hessian matrix of solutions is a generalization of the result of Kružkov [4]. On the other hand, our lower estimate on the Hessian matrix of solutions is best possible in some sense.

In this paper, we deal with the well-posedness issues of the generalized Benjamin-Ono-Burgers (gBOB) equations which are interpolated between the ordinary BOB equation and the KdV-Burgers equation with respect to the dispersive terms. We solve the initial-value problem (IVP) with data below $H ^{-1 /2}$, where $s = -1/2$ is the threshold for the well posedness of the Burgers equation. Our proof is based on the method by L. Molinet and F. Ribaud, which is analogous to that developed by J. Bourgain and C.E. Kenig, G. Ponce, and L. Vega. Interestingly, it is known that such a method cannot be applied to the Benjamin-Ono equation with initial data in $H ^s(\mathbb R)$, $s \in \mathbb R$.

The main purpose of this work is to further develop ideas and methods from a recent paper by the authors. In particular, we obtain new blow-up results for solutions of the inequality $$ |u|_t \geq \Delta [|u|^\sigma u] + |u|^{q} + \omega (x) $$ on the half-space ${\mathbb R}^1_+ \times {\mathbb R}^n$, where $n\geq 1$, $\sigma\geq 0$, $q>1+\sigma$, and $\omega: {\mathbb R}^n \to {\mathbb R}^1$ is a globally integrable function such that $\int_{{\mathbb R}^n} \omega (x) dx >0$, and establish that for $n>2$ the critical blow-up exponent $q^*=n(1+\sigma)/(n-2)$ is of the blow-up type.