An energy inequality and its applications of nonlocal boundary conditions of mixed problem for singular parabolic equations in nonclassical function spaces (Q1954341)

Summary: The aim of this paper is to establish a priori estimates of the following nonlocal boundary conditions mixed problem for parabolic equation: \(\partial v/\partial t - (a(t)/x^2)(\partial/\partial x)(x^2\partial v/\partial x) + b(x, t)v = g(x, t)\), \(v(x, 0) = \psi(x)\), \(0 \leq x \leq \ell\), \(v(\ell, t) = E(t)\), \(0 \leq t \leq T\), \(\int^{\ell}_0 x^3 v(x, t)dx = G(t)\), \(0 \leq t \leq \ell\). It is important to know that a priori estimates established in nonclassical function spaces is a necessary tool to prove the uniqueness of a strong solution of the studied problems.