| Abstract: |
| Abstract:
We study the local H\"{o}lder continuity of nonnegative solutions to the doubly nonlinear parabolic equation
$$ u_t-\textrm{div} \big(|Du^m|^{p-2} Du^m \big)=0 $$
in the mixed degenerate-singular cases, up to certain Barenblat numbers:
$$ 0\lt m(p-1)\lt1, \quad p+N(m(p-1)-1)>0, $$
or
$$ m(p-1)>1,\quad \frac{2N}{N+1}\lt p\lt 2, $$
where $N$ denotes the space dimension. The proof is based on the expansion of positivity and on a new version of integral Harnack-type inequalities, which are of independent interest. |
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