Frolenkov I.V.   Romanenko G.V.  

On existence of Cauchy problems solutions for two-dimensional loaded parabolic equations and systems of special form

Reporter: Romanenko G.V.

There are different methods using the overdetermination conditions to bring the inverse problem to the direct problem by the investigation of the coefficient inverse problems for parabolic equations (or systems) with the Cauchy data. One of them tells that the inverse problem is reduced to the non-classical direct problem for the loaded (containing traces of unknown functions and their derivatives) equation (or system of loaded equations). You have to know the conditions under which these auxiliary problems are solvable, as well as to know the properties of their solutions.

The sufficient conditions of the existence of solutions in classes of smooth bounded functions were found in the paper:

• the problem for two-dimensional loaded parabolic equation of special form (coefficients of the highest, lowest terms, and the right side depends on these unknown functions and their derivatives) with the Cauchy data;
• the Cauchy problem for the one-dimensional equation of the Burgers-type (a nonlinear equation for the function solutions of lower derivatives in the space variable);
• the problem for a system of two one-dimensional loaded parabolic equations with Cauchy data.

Bibliography:

1. Belov Yu.Ya. Inverse Problems for Partial Differential Equations. - Utrecht: VSP, 2002, 211 p.
2. Belov Yu.Ya, Korshun K.V. An Identification Problem of Source Function in the Burgers-type Equation // J. Sib. Fed. University. Math. Phys. 2012. V.5, N 4. P. 497–506.
3. Frolenkov I.V., Belov Yu.Ya. On existence of solutions for a class of two-dimensional loaded parabolic equations with Cauchy data // Non-classical equations of mathematical physics,  Pub. Institute of Math., Novosibirsk, 2012, P. 262-279.