On the critical Fujita exponents for solutions of quasilinear parabolic inequalities
This work is devoted to the study of critical blow-up phenomena for the inequality \[ u_t\geq A(u)+|u|^{q-1}- u\tag{1} \] in the half-space \(\mathbb{S}=\mathbb{R}^1_+\times \mathbb{R}^n\), \(n\geq 1\). This problem is considered for wide classes of quasilinear partial differential operators \(A(u)\) in divergent form. Well-known examples of such operators are (for \(\alpha>0)\) \[ A(u)= \text{div}_x\Bigl[ \biggl|\bigl( |u|^{m-1}\cdot u\bigr)_x\biggr |^{ \alpha-2} \cdot\bigl(|u|^{m-1} \cdot u\bigr)_x\Bigr],\;m\geq 1, \] \[ A(u) =\text{div}_x\biggl[ \bigl(|u|^{m-1}\cdot u\bigr)_x\biggr],\;m\geq 1,\;A(u)=\text{div}_x \bigl[|u_x|^{\alpha-2} \cdot u_x\bigr]. \] The authors obtain the existence of critical blow-up exponents for nonnegative weak solutions of the inequality (1) that belong only locally to the corresponding Sobolev spaces in the half-space \(\mathbb{S}\).
scientific article; zbMATH DE number 1764671