TY - JOUR
AU - Singer, Thomas
AB - We are interested in existence and regularity results for weak solutions of parabolic equations of the type\[\partial _t u- \operatorname {div}\,a(x,t,Du)=0\]on a parabolic space time cylinder $\Omega _T$. The vector field $a$ is assumed to satisfy a non-standard $p,q$-growth condition. We treat the subquadratic case, where\[\frac {2n}{n+2}\lt p\lt 2 \quad {\text {and}} \quad p\leq q \lt p+ \frac {4}{n+2}\]holds. We show existence of weak s$u \in L^p(0,T;W^{1,p}(\Omega )) \cap L^q_{\mathrm {loc}}(0,T;W^{1,q}_{\mathrm {loc}}(\Omega ))$ for the Cauchy–Dirichlet problem associated to the parabolic equation from above. Further, a local bound for the spatial gradient $Du$ is established. The results cover for example equations of the type\[\partial _t u = \operatorname {div} (\alpha (x,t) (\mu ^2+ |Du|^2)^{(p-2)/2}Du) + \operatorname {div}(\beta (x,t)(\mu ^2 + |Du|^2)^{(q-2)/2}Du)\]with $\mu \in [0,1]$ and suitable functions $\alpha (x,t)$ and $\beta (x,t)$. We emphasize that the results cover the singular case $\mu =0$.
TI - PARABOLIC EQUATIONS WITH p,q-GROWTH: THE SUBQUADRATIC CASE
JF - The Quarterly Journal of Mathematics
DO - 10.1093/qmath/hav005
DA - 2015-06-19
UR - https://www.deepdyve.com/lp/oxford-university-press/parabolic-equations-with-p-q-growth-the-subquadratic-case-ucTYOF7O3e
SP - 707
EP - 742
VL - 66
IS - 2
DP - DeepDyve
ER -