A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li &amp; Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in ^{[<xref ref-type="bibr" rid="b5">5</xref>]} for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains

<p style='text-indent:20px;'>We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in <i>C</i>¹ and <i>C</i>^{<i>k</i>, <i>α</i>} domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.</p><p style='text-indent:20px;'>As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.</p>

On the weak Harnack inequality for the parabolic p ( x )-Laplacian

In this paper a weak Harnack inequality for the parabolic p ( x )-Laplacian is established.

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

In this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.