In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.

We study a class of third order hyperbolic operators $P$ in $G=\Omega \cap \{0\le t\le T\},\Omega \subset {ℝ}^{n+1}$ with triple characteristics on $t=0$. We consider the case when the fundamental matrix of the principal symbol for $t=0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t\>0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P+Q$ is well posed in $G$ for any lower order terms $Q$.

The existence, uniqueness and regularity of the generalized local solution and the classical local solution to the periodic boundary value problem and Cauchy problem for the multidimensional coupled system of a nonlinear complex Schrödinger equation and a generalized IMBq equation $$i{\epsilon }_t+{\nabla }^2\epsilon -u\epsilon =0,$$$$u_{...}$$

We study local and global Cauchy problems for the Semilinear Parabolic Equations ∂tU - ΔU = P(D) F(U) with initial data in fractional Sobolev spaces Hps(Rn). In most of the studies on this subject, the initial data U0(x) belongs to Lebesgue spaces Lp(Rn) or to supercritical fractional Sobolev spaces Hps(Rn) (s > n/p). Our purpose is to study the intermediate cases (namely for 0 < s < n/p). We give some mapping properties for functions with polynomial growth on subcritical Hps(Rn)...

This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $${\displaystyle \frac{\partial u}{\partial t}}-(\lambda +\mathrm{i}\alpha )\Delta u+(\kappa +\mathrm{i}\beta ){\left|u\right|}^{q-1}u-\gamma u=0$$ in ${ℝ}^N\times (0,\infty )$ with $L^p$-initial data $u_0$ in the subcritical case ($1\le q<1+2p/N$), where $u$ is a complex-valued unknown function, $\alpha$, $\beta$, $\gamma$, $\kappa \in ℝ$, $\lambda >0$, $p>1$, $\mathrm{i}=\sqrt{-1}$ and $N\in \mathbb{N}$. The proof is based on the $L^p$-$L^q$ estimates of the linear semigroup $\{exp(t(\lambda +\mathrm{i}\alpha )\Delta \left)\right\}$ and usual fixed-point argument.

We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor ${\tau }^V\left(𝕖\right)=\tau \left(𝕖\right)-2{\mu }_1\Delta 𝕖$, where the nonlinear function $\tau \left(𝕖\right)$ satisfies ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge c{\left|𝕖\right|}^p$ or ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge {c\left(\right|𝕖|}^2+{\left|𝕖\right|}^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p>1$ for both models. Then, under vanishing higher viscosity ${\mu }_1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p>\frac{3n}{n+2}$, its uniqueness and...

In this work we present a class of partial differential operators with constant coefficients, called multi-quasi-hyperbolic and defined in terms of a complete polyhedron. For them we obtain the well-posedness of the Cauchy problem in generalized Gevrey classes determined by means of the same polyhedron. We present some necessary and sufficient conditions on the operator in order to be multi-quasi-hyperbolic and give some examples.

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ${ℝ}^n$ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ${ℝ}^n$, $n\in \mathbb{N}$. In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Let $Q_T$ be a cylinder in ${\mathbb{R}}^{n+1}$ and $x=\left(x^{\prime },t\right)\in {\mathbb{R}}^n\times \mathbb{R}$. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator $$\left\{\begin{array}{cc}{u}_t-\stackrel{n}{\sum _{i,j=1}}{a}^{ij}\left(x\right)D_{ij}u=f\left(x\right)\hfill & \text{q.o. in }{Q}_T,\hfill \\ u\left(x\right)=0\hfill & \text{su }\partial Q_T,\hfill \end{array}\right.$$ in the Morrey spaces $W_{p,\lambda }^{2,1}\left(Q_T\right)$, $p\in \left(1,\mathrm{\infty }\right)$, $\lambda \in \left(0,n+2\right)$, supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.

We prove local in time solvability of the nonlinear initial-boundary problem to nonlinear nondiagonal parabolic systems of equations (multidimensional case). No growth restrictions are assumed on generating the system functions. In the case of two spatial variables we construct the global in time solution to the Cauchy-Neumann problem for a class of nondiagonal parabolic systems. The solution is smooth almost everywhere and has an at most finite number of singular points.