Of concern is the nonlinear evolution equation \begin{align} \frac{du}{dt} & \in A(t)u,\ \ 0 < t < T \notag \\ u(0) & = u_{0} \notag \end{align} in a real Banach space $ X $, where $ A(t) : D(A(t)) \subset X \longrightarrow X $ is a time-dependent, nonlinear, multivalued operator acting on $ X $. It is shown that under certain assumptions on $ A(t) $, the equation has a strong solution. Applications to nonlinear parabolic boundary value problems with time-dependent boundary conditions are given.

We study the blow-up and stability of solutions of the equation $u_t+u_x=u_{xx}+{\lambda} f(u)/ (\int^1_0 f(u)\,dx )^2$ with certain initial and boundary conditions. When $f$ is a decreasing function, we show that if $\int^{\infty}_0f(s)\,ds <\infty$, then there exists a ${\lambda}^{*}>0$ such that for ${\lambda}>{\lambda}^{*}$, or for any $0 <{\lambda}\leq {\lambda}^{*}$ but with initial data sufficiently large, the solutions blow up in finite time. If $\int^{\infty}_0f(s)\,ds=\infty$, then the solutions are global in time. The stability of solutions in both cases is discussed. We also study the case of $f$ being increasing.

In this paper pencils of partial differential operators depending polynomially on a complex parameter and corresponding boundary value problems with general boundary conditions are studied. We define a concept of ellipticity for such problems (for which the parameter-dependent symbol in general is not quasi-homogeneous) in terms of the Newton polygon and introduce related parameter-dependent norms. It is shown that this type of ellipticity leads to unique solvability of the boundary value problem and to two-sided a priori estimates for the solution.

Evolution equations are considered as operator equations involving a sum of the time-derivative operator $ \partial _{0} $ regarded as a normal operator in a suitable Hilbert space setting and another fairly arbitrary spatial operator $ A$ acting in a Hilbert space $ H $. As an extension of the linear theory the case of a class of nonlinear constitutive laws with memory effects is considered.

We improve results in reference [6] concerning the effect of the direction of the vorticity on the regularity of weak solutions to the 3D Navier--Stokes equations. In particular, we prove that, if the direction of the vorticity belongs to suitable Sobolev spaces, then there exists a unique smooth solution of the Cauchy problem for the Navier--Stokes equations.

In this work, we prove some "precise properties" for radially symmetric supersolutions for polyharmonic operators with zero Dirichlet boundary conditions. As a consequence, we prove that they are strictly monotone functions of the radius.

This paper studies the large-time behavior of small solutions to the nonlinear Schrödinger equations in one space dimension. Our relevant equations contain the gauge-invariant cubic nonlinearities of derivative type. Since the nonlinear term is the so-called long-range type, it is well-known that the nonlinear solution tends to the modified linear solution called the first asymptotic term. We present the higher-order asymptotic expansion of the nonlinear solution in weighted $L^2$ and $L^{\infty}$ spaces. The result shows that the nonlinear interaction plays an explicit role in the higher-order asymptotic terms as well as in the phase modification. Our method relies on the nonlinear gauge transformations and the application of $L^{\infty}$ decay estimate by Hayashi--Naumkin [13, 12] for estimating the nonlinear solution in Sobolev spaces.