A PRIORI ESTIMATES, EXISTENCE THEOREMS, AND THE BEHAVIOR AT INFINITY OF SOLUTIONS OF QUASIELLIPTIC EQUATIONS IN $ \mathbf{R}^n$

Solutions of the heat equation in domains with singularities

“…1 ~= 2~ ~ 144 on the interval 0 < ~ < a in (14). Then the equation becomes in (14) on the interval -fl < ~ < 0, we obtain the equation…”

“…(1) and the right-hand side can be expanded in asymptotic series in powers of r near the singular point FI, then in the same way as in [3] we can obtain the expansion of the solution in powers of r. Equation (4) For p ~ 0, the transformation ~b is infinitely differentiable and its Jacobian does not vanish [14]. This transformation takes K to f~0 • (0, co), where f~0 = f~ N K. In Eq.…”

“…For y~(~) and Y2(~) we obtain equations of the form (14), but there is only one boundary condition for each of these equations, namely, yl(-a) = 0 and y2(-/~) = 0, respectively. The general solution of each of these equations has the following form (the derivation is the same as in the domain K1 ):…”

Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

“…Obviously, for p # 0 the transformation r is smooth and its Jacobian is nondegenerate [12]. We obtain the estimate (6) by multiplying this by 2 "d and then by summing over all j.…”

The Differential Properties of Functions from Sobolev-Morrey Type Spaces of Fractional Order

“…In (L. Arkeryd, 1969), L p − estimates for solutions were studied, under the condition that the coefficients of leading derivatives are infinitely differentiable, and in (L. A. Bagirov, 1979;S. V. Uspenskii, G. V. Demidenko and V. G. Perepelkin, 1984) some other problems of the theory of quasielliptic equations were considered.…”