Quasilinear Parabolic Equations in Divergence Form with Singular Coefficients: The \(p\)-Gradient-Form-Boundedness Approach
Applicable Nonlinear Analysis, Volume 4, Issue 1, April 2027, Pages: 67–88
Mykola Yaremenko
National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, 37, Prospect Beresteiskyi (former Peremohy), Kyiv, Ukraine, 03056
Abstract
We study the initial-boundary value problem for a class of quasilinear parabolic equations in divergence form with natural growth of the lower-order term and singular coefficients. The structural assumptions are those of the Ladyzhenskaya-Solonnikov-Uraltseva \(pk\)-class, except that the classical anisotropic integrability condition on the gradient coefficient is replaced by a \(p\)-gradient-form-boundedness condition tailored to the energy structure. This condition contains the classical \(L^r\) case (\(r>n+2\)) as a special instance, while also allowing coefficients with weak-\(L^{n+2}\) singularities. Under these assumptions we give a self-contained proof of local and global a priori estimates: a Caccioppoli-type inequality, local boundedness via De Giorgi–Ladyzhenskaya iteration, and a global energy bound under a smallness condition. Existence of weak solutions is obtained by the Galerkin method combined with the Minty–Browder technique for strong convergence of gradients. Maximum and comparison principles are established, and the Hölder continuity of bounded weak solutions is proved both in the interior and up to the boundary for Lipschitz cylinders. The results hold on arbitrary bounded space-time cylinders without any regularity hypothesis on the boundary. The paper systematically develops the \(p\)-gradient-form-boundedness framework and shows that the essential tools of the De Giorgi–Ladyzhenskaya theory remain valid under this weaker condition.
Cite this Article as
Mykola Yaremenko, Quasilinear Parabolic Equations in Divergence Form with Singular Coefficients: The \(p\)-Gradient-Form-Boundedness Approach, Applicable Nonlinear Analysis, 4(1), 67–88, 2027