Parabolic Equations with p, q-Growth: the Subquadratic Case

singer, thomas;

We are interested in existence and regularity results for weak solutions of parabolic equations of the type ∂tu − div a(x, t, Du) = 0
on a parabolic space time cylinder T . The vector field a is assumed to satisfy a non-standard p, q-growth condition. We treat the subquadratic case, where 2n/n + 2 < p <2 and p ≤ q < p + 4/n + 2
holds. We show existence of weak su ∈ Lp(0, T;W1,p()) ∩ Lq
loc(0, T;W1,q loc ()) 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 ∂tu = div(α(x, t)(μ2 + |Du|2)(p−2)/2Du) + div(β(x, t)(μ2 + |Du|2)(q−2)/2Du) with μ ∈ [0, 1] and suitable functions α(x, t) and β(x, t). We emphasize that the results cover the
singular case μ = 0.