We use maximal $L_p$~regularity to study quasilinear parabolic evolution equations. In contrast to all previous work we only assume that the nonlinearities are defined on the space in which the solution is sought for. It is shown that there exists a unique maximal solution depending continuously on all data, and criteria for global existence are given as well. These general results possess numerous applications, some of which will be discussed in separate publications.

A scalar-field type equation with the Robin condition is discussed. For some range of the parameter, the multiplicity of solutions is obtained. Moreover, one of the obtained solutions has a blowup property and the blowup point is determined.

Under suitable conditions on the functions $a\in C^1({\mathbb R}^N,{\mathbb R}^{N^2})$, $F\in C^1({\mathbb R}^N,{\mathbb R}^N)$, and $V:{\mathbb R}^N\to [0,\infty)$, we show that the operator $Au=\nabla (a\nabla u) +F\cdot \nabla u-Vu$ with domain $W^{2,p}_V({\mathbb R}^N)= \{ u\in W^{2,p}({\mathbb R}^N):Vu\in L^p({\mathbb R}^N) \}$ generates a positive analytic semigroup on $L^p({\mathbb R}^N)$, $1 < p < \infty$. Analogous results are also established in the spaces $L^1({\mathbb R}^N)$ and $C_0({\mathbb R}^N)$. As an application we show that the generalized Ornstein--Uhlenbeck operator $A_{\Phi,G} u=\Delta u -\nabla \Phi \cdot \nabla u + G\cdot \nabla u$ with domain $W^{2,p}({\mathbb R}^N,\mu)$ generates an analytic semigroup on the weighted space $L^p({\mathbb R}^N,\mu)$, where $1 < p < \infty$ and $\mu(dx)=e^{-\Phi(x)}dx$.

We consider the Cauchy problem \[\left\{ \begin{array}{lll} P\bigl(t,x,D^{m-1}u,D_{t},D_{x}\bigr)u(t,x)=f(t,x,D^{m-1}u) \\ \partial^j_{t} u(0,x)=u_{j}(x),\ \ \ j=0,...,m-1, \end{array} \right.\] in $[0,T]\times{\mathbb{R}}^n$ for a quasilinear weakly hyperbolic operator \[ P\bigl(t,x,D^{m-1}u,D_{t},D_{x}\bigr)=D_{t}^{m}+\sum_{j=0}^{m-1} \sum_{|\alpha|={m-j}}a_{\alpha}^{(j)}(t,x,D^{m-1}u)D_x^{\alpha}D_t^j\] with coefficients $a_{\alpha}^{(j)}$ having the first time derivative with singular behavior of the type $t^{-q}$, $q>1$, as $t\to 0$. We show that for $t\leq T_0^\ast$, $T_0^\ast$ sufficiently small, given Cauchy data in a Gevrey class $G^\sigma$ there exists a unique solution $u\in C^{m-1}([0,T_0^\ast];G^\sigma)$ provided that $\sigma < \frac{qr}{qr-1}$ where $r$ denotes the largest multiplicity of the characteristic roots.

In this paper we study infinite horizon variational problems arising in continuum mechanics and establish a generic uniqueness of an optimal solution.