Existence results for a Cauchy–Dirichlet parabolic problem with a repulsive gradient term

Abstract

ABSTRACT. We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following:in Ω,

Local and global time decay for parabolic equations with super linear first-order terms

“…For the interested reader, let us stress that existence results for more general operators (including the ‐Laplacian) and general right‐hand sides can be found in . The present study of short and long time decay can also be extended to the ‐Laplace operator with similar techniques, but a similar development would have resulted in a too long exposition.…”

“…This particular value of implies that . However, ‐data are not admissible for a general solvability in this case (see also ). Therefore, in the class of Lebesgue spaces we should take into account initial data belonging to for some .…”

“…We also postpone the proof of this result to the Appendix. Let us recall that the existence of at least one solution satisfying () (if ) or () (if ) is proved in .…”

Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

“…for any µ > 0, where we have used that z ε n (0) ≤ v 0 . Now, reasoning as in the proof of the a priori estimates contained in [23]. We fix a value δ 0 such that max δ L r (0,T ;L m (Ω)) < δ 0 for any k ≥ k 0 .…”

“…Proof. We proceed observing that, thanks to (A.3)-(A.4), then t − N 2σ U R √ t ′ ∈ C 1 ([0, T ]) and thus (A.6) admits a solution v 1 such that v 1 ∈ C([0, T ]; L σ (Ω)) and |v 1 | σ 2 ∈ L 2 (0, T ; H 1 0 (B R (0))) (see [23]). Then, we are left with the proofs of (A.7)-(A.8).…”