A fundamental solution of the parabolic equation on Hilbert space. II. The semigroup property

This paper concerns solutions with time-dependent singularities for a semilinear parabolic equation with a superlinear absorption term. Here, by time-dependent singularity, we mean a singularity with respect to the space variable whose position depends on time. It is shown that if the power of the nonlinearity is in some range, then any singularity is removable. On the other hand, in other range, two types of time-dependent singular solutions exist: One resembles the fundamental solution of the Laplace equation near the singular point, and the other has a stronger singularity.

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THE VALUE OF THE SOLUTION OF ONE PROBLEM FOR THE PSEUDOPARABOLIC EQUATION IN THE CLASS OF GELS

Bulletin Series of Physics & Mathematical Sciences ◽

10.51889/2020-2.1728-7901.11 ◽

2020 ◽

Vol 70 (2) ◽

pp. 77-83

Author(s):

U.K. Koylyshov ◽

◽

A.Zh. Aldashova ◽

Keyword(s):

Cauchy Problem ◽

Parabolic Equation ◽

Fundamental Solution ◽

A Priori Estimates ◽

Dimensional Space ◽

Three Dimensional ◽

Initial Condition ◽

Pseudoparabolic Equation ◽

Volume Potential ◽

The Cauchy Problem

This article discusses the Cauchy problem for a pseudo-parabolic equation in three-dimensional space. The result can be generalized to - dimensional space. The Cauchy problem for equations of parabolic and elliptic types is well studied. For a pseudo-parabolic equation using the previously constructed fundamental solution, evaluating the fundamental solution and its derivatives. Applying the Fourier transform with respect to and the Laplace transform with, we first obtained a priori estimates for the potentials of the initial condition and the volume potential in Hölder spaces. Further, using these results, we have proved an estimate of the solution of the Cauchy problem for the pseudo-parabolic equation in Hölder classes. A detailed proof of the estimation of the potentials of the initial condition, the volume potential, and the solution of the Cauchy problem for the pseudoparabolic equation is given

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Fundamental solution for a one-dimensional parabolic equation with a continuous coefficient

Proceeding of the International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin ◽

10.4213/proc22958 ◽

2018 ◽

Author(s):

Elena Aleksandrovna Baderko ◽

K V Semenov

Keyword(s):

Parabolic Equation ◽

Fundamental Solution ◽

One Dimensional ◽

Continuous Coefficient

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ON THE CONVERGENCE AND RATE OF THE CONVERGENCE OF A PROJECTION-DIFFERENCE METHOD FOR APPROXIMATE SOLVING A PARABOLIC EQUATION WITH WEIGHT INTEGRAL CONDITION

Tambov University Reports Series Natural and Technical Sciences ◽

10.20310/1810-0198-2018-23-123-517-523 ◽

2018 ◽

pp. 517-523

Author(s):

Anastasiya Alexandrovna Petrova

Keyword(s):

Finite Element Method ◽

Hilbert Space ◽

Approximate Solution ◽

Parabolic Equation ◽

Integral Condition ◽

Approximate Solutions ◽

Linear Parabolic Equation ◽

The Finite Element Method ◽

Spatial Variables ◽

Euler’S Method

In the Hilbert space the abstract linear parabolic equation with nonlocal weight integral condition for the solution is resolved approximately by projectiondifference method using time-implicit Euler’s method. Approximation of the problem by spatial variables is oriented on the finite element method. Errors estimations of approximate solutions, convergence of approximate solution to exact one and orders of rate of convergence are established.

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