Solvability Of Parabolic Equations Research Articles

On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators

On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators

Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

We prove the \(L_{p,q}\)-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

On Coercive Solvability of Parabolic Equations with Variable Operators

In a Banach space E, the Cauchy problem $$ \upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\kern1em \upsilon (0)={\upsilon}_0, $$ is considered for a differential equation with linear strongly positive operator A(t) such that its domain D = D(A(t)) does not depend on t and is everywhere dense in E and A(t) generates an analytic semigroup exp{−sA(t)}(s ≥ 0). Under natural assumptions on A(t), we prove the coercive solvability of the Cauchy problem in the Banach space $$ {C}_0^{\beta, \upgamma} $$ (E). We prove a stronger estimate for the solution compared with estimates known earlier, using weaker restrictions on f(t) and v0.

О первой смешанной задаче в банаховых пространствах для вырождающихся параболических уравнений с меняющимся направлением времени

The article is devoted to studying one of the sections of nonclassical differential equations, namely, matters concerned with solvability of parabolic equations with changing second-order time direction. As is known, in ordinary boundary-value problems for strictly parabolic equations, the smoothness of the initial and boundary conditions completely ensures that the solutions belong to the Holder spaces, but in the case of equations with changing time direction, the smoothness of the initial and boundary conditions does not ensure that the solutions belong to these spaces. S.A. Tersenov (for a model parabolic equation with changing time direction) and S.G. Pyatkov (for a more general second-order equation) obtained the necessary and sufficient conditions for solvability of the corresponding mixed problems in Holder spaces. In so doing, they always assumed the initial and boundary conditions being equal to zero. Cases in which the initial and boundary conditions belong to Banach spaces are considered. The functional spaces in which the solutions must be sought are introduced. Relevant a priori estimates, which make it possible to obtain the solvability conditions for these problems, are obtained. The properties of the obtained solutions have been studied. In particular, the equivalence of the Riesz and Littlewood-Paley conditions similar to the conditions for solutions of strictly elliptic and strictly parabolic second order equations is established. A unique solvability of the first mixed problem with boundary and initial functions from the Banach space has been proved.

Solvability of parabolic equations in divergence form with partially BMO coefficients

We prove the H p 1 solvability of second order parabolic equations in divergence form with leading coefficients a i j measurable in ( t , x 1 ) and having small BMO (bounded mean oscillation) semi-norms in the other variables. Additionally we assume a 11 is measurable in x 1 and has small BMO semi-norms in the other variables. The corresponding results for the Cauchy problem are also established. Parabolic equations in Sobolev spaces H q , p 1 with mixed norms are also considered under the same conditions of the coefficients.

Unique Solvability of Parabolic Equations with Almost-Periodic Coefficients in Hölder Spaces

We obtain a criterion for invertibility in Holder spaces of linear parabolic operators of arbitrary order with any number of spatial variables and almost-periodic coefficients.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.