Supersolutions for a class of nonlinear parabolic systems

Abstract

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t v=\Delta v+u^q, & x\in\Omega,\,\,\,t>0,\\ u=v=0, & x\in\partial\Omega,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\in\Omega, \end{array} \right. $$ where $p\ge 0$, $q\ge 0$, $\Omega$ is a (possibly unbounded) smooth domain in ${\bf R}^N$ and both $u_0$ and $v_0$ are nonnegative and locally integrable functions in $\Omega$. The supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow-up rate of the solutions

Solvability of the Heat Equation with a Nonlinear Boundary Condition

“…which together with (4.2) implies that sup 0<τ <t/2 D\B(0,nt We prove Theorems 1.3, 1.4 and 1.5 by modifying the arguments in [8,10,14]. Lemma 5.1 is a key lemma in our proofs.…”

“…Furthermore, they showed that, if problem (1.3) possesses a local-in-time nonnegative solution, then its initial trace µ satisfies the following: Here q θ := 1 + θ/N . In [8], developing the arguments in [10] and [14], they also obtained sufficient conditions on the initial data for the existence of the solution of (1.3) and identified the strongest singularity of the initial data for which the Cauchy problem to (1.3) possesses a local-in-time nonnegative solution.…”

Existence of solutions for a fractional semilinear parabolic equation with singular initial data

“…It follows from (3.32) that We modify the arguments in [11] and [17] to prove Theorems 1.3, 1.4 and 1.5. In the rest of this paper, for any two nonnegative functions f 1 and f 2 defined in a subset D of [0, ∞),…”

“…On the other hand, Takahashi [21] recently proved that, in the case p ≥ p * , for any γ > 0, Cauchy problem (1.2) The local solvability of Cauchy problem (1.2) has been studied in many papers (see e.g., [1,2,5,7,10,11,13,17,18,19,21,22,23] and references therein). It is known that there exists a constant c 2 > 0 such that Cauchy problem (1.2) possesses a solution in R N × [0, ρ 2 ], where ρ > 0, if p > p * and sup x∈R N µ L r,∞ (B(x,ρ)) ≤ c 2 with r = N (p − 1) 2 (1.3) (see [11]). See also [10,13,17].…”

Thresholds on growth of nonlinearities and singularity of initial functions for semilinear heat equations

“…We introduce a simple but new supersolution (3.5), using J. In [8,10,20] similar functions were also used as supersolutions. However, these supersolutions were directly related to integrability conditions.…”