The Cauchy problem for a system of quasilinear parabolic equations of chemical kinetics type

Abstract

The fundamental questions of the theory of quasilinear parabolic systems are the questions of the solvability of boundary value problems and of the Cauchy problems for these systems, and also the questions related to the asymptotic behavior of the solutions when the time increases unboundedly. Numerous investigations on the qualitative theory of these systems have been inspired by the well-known classical work of A. N. Kolmogorov, I. G. Petrovskii, N. S. Piskunov (1937). A large theoretical and applied interest has the investigation of systems that arise in the mathematical modeling of chemical processes, the so-called reaction--diffusion systems.This paper is devoted to the investigation in the half-space

Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

“…Remark 4.21. The estimate in Theorem 4.16 was stated for semiconvex solutions of some stationary equations in Theorem 4 of [44], eq. ( 16), in L 1 norms with order O(ε ν ), ν ∈ (0, 1), and in eq.…”

“…( 16), in L 1 norms with order O(ε ν ), ν ∈ (0, 1), and in eq. ( 17) of [44] in L p norms for p sufficiently large, with order O(ε). The L 1 rate of order O(ε) for semiconcave solutions is the subject of Lemma 2 in [41] for uniformly convex H.…”

“…Theorem 5.5. Let u be a solution of (44) with H(0) = 0 and satisfying the one-side decay condition u(x, t) ≥ −|x|µ(|x|) + K(t), µ(r) ≥ 0, µ(r) → 0 as r → ∞, for a bounded function K. Then u must be constant.…”

“…The proof relies on showing that in the long-time regime the solution of (44) becomes concave. We can view a solution to (44) as a solution of the problem on the layer {−T ≤ t ≤ 0} by letting T → ∞ with initial condition u(−T ) = u(x, −T ). By Theorem 3.12 we have Proof.…”