Localized Boundary Blow-up Regimes for General Quasilinear Divergent Parabolic Equations of Arbitrary Order

A mixed nonhomogeneous Cauchy–Dirichlet problem is considered for a general quasilinear parabolic equation in the divergence form in the case when the boundary data have an unbounded blow-up at a finite moment $T$. The energy space of this equation is $L_{\infty ,\mathrm {loc}}(0,T;L_{q+1}(\Omega ))\cap L_{p+1,\mathrm {loc}}(0,T;W_{p+1}^m(\Omega ))$, $m\ge 1$, $p>q>0$. The asymptotic behavior of an arbitrary energy solution for $t\to T$ is studied. Sharp (in a sense) integral constraints are established for the blow-up rate of the boundary data which guarantee the localization of the singularity zone of a solution in a certain neighborhood of the boundary of a domain (S-regime) or on the boundary itself (LS-regime).