Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy...

We investigate existence and unicity of global sectorial holomorphic solutions of functional linear partial differential equations in some Gevrey spaces. A version of the Cauchy-Kowalevskaya theorem for some linear partial $q$-difference-differential equations is also presented.

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front.

The present paper deals with the numerical solution of the nonlinear heat equation. An iterative method is suggested in which the iterations are obtained by solving linear heat equation. The convergence of the method is proved under very natural conditions on given input data of the original problem. Further, questions of convergence of the Galerkin method applied to the original equation as well as to the linear equations in the above mentioned iterative method are studied.

In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧$-(a+b{\int }_{\Omega }\left|\nabla u\right|²dx)\Delta u=u⁵+\lambda u^{q-1}/{\left|x\right|}^{\beta }$ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

We study the microlocal analyticity of solutions $u$ of the nonlinear equation$$u_t=f(x,t,u,u_x)$$where $f(x,t,{\zeta }_0,\zeta )$ is complex-valued, real analytic in all its arguments and holomorphic in $({\zeta }_0,\zeta )$. We show that if the function $u$ is a $C^2$ solution, $\sigma \in \text{Char}{L}^u$ and $\frac{1}{i}\sigma \left([L^u,\overline{L^u}]\right)\&lt;0$ or if $u$ is a $C^3$ solution, $\sigma \in \text{Char}{L}^u$, $\sigma \left([L^u,\overline{L^u}]\right)=0$, and $\sigma \left([L^u,[L^u,\overline{L^u}]]\right)\ne 0$, then $\sigma \notin W{F}_{a}u$. Here $W{F}_{a}u$ denotes the analytic wave-front set of $u$ and Char$L^u$ is the characteristic set of the linearized operator. When $m=1$, we prove a more general result involving the repeated brackets of $L^u$ and $\overline{L^u}$ of any order.

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the traction at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and uniqueness of weak solutions (the latter for small data) and discuss particular applications of the results.