On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

Sufficient conditions are found for the correct and unique solvability of a class of third-order parabolic operator differential equations, whose principal parts have multiple characteristics, in a Sobolev-type space with exponential weight. The estimates for the norms of intermediate derivative operators are obtained and the relationship between these estimates and solvability conditions is established. Besides, the connection is found between the order of exponential weight and the lower bound for the spectrum of abstract operator appearing in the principal part of the equation.

Note that throughout this paper all the derivatives are understood in the sense of the theory of distributions, and the operator   is defined by the spectral decomposition of the operator ; that is,   = ∫ +∞      ,  ≥ 0, where   is a decomposition of the unit of the operator .Now, let us recall one fact related to the space  3 2 (; ).It is known that if V() ∈  3  2 (; ), then the inequalities           3−   V            2 (;) ≤   ‖V‖  3 2 (;) ,  = 0, 3, are valid, where   ,  = 0, 3, are constants independent of function V().This fact is referred to as the intermediate derivatives theorem (see [2,Chapter 1]).Also, these inequalities are usually referred to as Kolmogorov-type inequalities.Let −∞ <  < +∞.For the functions () defined on  with the values in , we introduce the following spaces with the weight  −(/2) : Obviously, in case  = 0 we get the spaces  2,0 (; ) =  2 (; ),  3 2,0 (; ) =  3 2 (; ).In the sequel, by (, ) we will mean a set of linear bounded operators from the Hilbert space  to another Hilbert space .If  = , we will write () instead of (, ).By () we will denote the spectrum of the operator .
Consider the operator differential equation where  =  * ≥ ,  > 0, then this vector function is called a regular solution of (5), and ( 5) is said to be regularly solvable.
The principal part of (5) has multiple characteristics, so, according to the classification of [3], this equation belongs to the class of parabolic operator differential equations.The equations of form ( 5) characterize the problems of diffusion or heat conductivity in viscoelastic media [4].Besides, such equations are also interesting in view of the fact that some classes of equations, which can be useful in modeling the problems of world population growth, can be reduced to them [5].
In this work, we find the regular solvability conditions for (5) on the entire axis.We also obtain the estimates for the norms of intermediate derivative operators in a Sobolevtype space through the norm of the operator generated by the principal part of (5) (it should be noted here that the estimates for the norms of intermediate derivatives for scalar functions have been obtained in [33,34] and the references therein).Moreover, we establish the relationship between these estimates and the regular solvability conditions for (5).The found regular solvability conditions are expressed in terms of operator coefficients of (5), which makes them easily verifiable and convenient for use both in theoretical problems and in applications. 3 2, (;)
As is known, the intermediate derivative operators are continuous [2].By virtue of this fact and Corollary 3, the norms of the operators (30) can be estimated through ‖ 0 ‖  2, (;) .
The lemma below can be proved with the help of Theorem 2 and Lemma 6. be true, where the numbers  1 (;  0 ) and  2 (;  0 ) are defined as in Theorem 5. Then ( 5) is regularly solvable.