Nonnegative Classical Solutions Research Articles

Conservation of boundary decay and nonconvergent bounded gradients in degenerate diffusion problems

This paper investigates the boundary behavior of nonnegative classical solutions to the Dirichlet problem for $$ u_t=u^p\Delta u + g(u) \qquad \mbox{in } \Omega \times (0,T), \qquad p>1, $$ and draws some consequences for the large time behavior of solutions. Here, $\Omega \subset \mathbb R^n$ is a smooth bounded domain and $g:\ [0,\infty) \to \mathbb R$ is locally Lipschitz continuous with $g(0)=0$. The first goal is to study for which $\alpha>0$ the implication \begin{align} & & u_0(x) \le c_1 ({{\rm dist} \, } (x,{\partial\Omega}))^\alpha \qquad (c_1>0) \notag \\ & \Rightarrow & u(x,t) \le C(T') ({{\rm dist} \, }(x,{\partial\Omega}))^\alpha \quad \mbox{in } \Omega\times (0,T') \ \mbox{for any } T' < T, \qquad \mbox{(I)} \tag*{(I)} \end{align} is valid, and it turns out that this holds whenever either $p \ge 2$, or $p < 2$ and $\alpha \ge \frac{1}{p-1}$. For $p \in (1,2]$ and $g\equiv 0$, this complements a previously known result, according to which the lower estimate $u_0(x)\ge c_0(\text{dist}\, (x,{\partial\Omega}))^\alpha$ with some $\alpha <\frac{1}{p-1}$ and $c_0>0$ implies the existence of $T>0$ and $C>0$ such that $u(x,t) \ge C \text{dist}\, (x,{\partial\Omega})$ for all $x \in \Omega$ and $t \ge T$. Using (I), we moreover show that whenever $p>1$, there exist some values of $q \ge 1$ such that the particular equation, $u_t=u^p u_{xx}+u^q$, possesses positive classical solutions which are nondecreasing w.r.~to $t$ and remain uniformly bounded in $C^1(\bar\Omega)$ for all times, but do not converge in $C^1(\bar\Omega)$ as $t\to\infty$.

Finite time vs. infinite time gradient blow-up in a degenerate diffusion equation

This paper deals with the phenomenon of gradient blow-up of nonnegative classical solutions of the Dirichlet problem for (*) u t = u p u xx + Ku r u x 2 + u q in Ω x (0,T) in a bounded interval Ω c R, where p > 2, 1 ≤ q ≤ p- 1, r ≥ 1, K ≥ 0, and the initial data u o are assumed to belong to C 1 (Q) and satisfy u 0 (x) > C 0 dist(x, δΩ) in Q with some C 0 > 0. It is shown that if the gradient term in (*) is weak enough near u = 0, then all bounded solutions undergo an infinite time gradient blow-up, whereas if this term is sufficiently strong, then all solutions blow up in C 1 (Ω) within finite time. Here by 'weak' we mean that the parameters satisfy either r = p - 1 and K p - 1, and by 'strong' the precise opposite, that is, either r = p - 1 and K > p - 2, or r < p - 1.

Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations

In this paper, we study some new connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to superlinear parabolic problems, with or without boundary conditions. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of solutions from Liouville-type theorems, which unifies and improves many results concerning a priori bounds, decay estimates and initial and final blow-up rates. For example, for the equation u t - Δu = u p on a domain Ω, possibly unbounded and not necessarily convex, we obtain initial and final blow-up rate estimates of the form u(x, t) ≤ C(Ω, p)(1 + t -1/(p-1) + (T - t) -1/(p-1) ). Our method is based on rescaling arguments combined with a key doubling property, and it is facilitated by parabolic Liouville-type theorems for the whole space or the half-space. As an application of our universal estimates, we prove a nonuniqueness result for an initial boundary value problem.

Global existence and blow up for a parabolic system with nonlinear boundary condition

In this paper the nonnegative classical solutions of a parabolic system with nonlinear boundary conditions are discussed. The existence and uniqueness of a nonnegative classical solution are proved. And some sufficient conditions to ensure the global existence and nonexistence of nonnegative classical solution to this problem are given.

Extinction and decay estimates for viscous Hamilton-Jacobi equations in ℝ^{ℕ}

We consider non-negative and integrable classical solutions to the Cauchy problem u t − Δ u + | ∇ u | p = 0 u_t-\Delta u+\vert \nabla u\vert ^p=0 when p ∈ ( 0 , + ∞ ) p\in (0,+\infty ) . For p ∈ ( 0 , N / ( N + 1 ) ) p\in (0,N/(N+1)) we prove that any such solution vanishes identically after a finite time. For higher values of p p temporal decay estimates are obtained.

A semilinear parabolic system in a bounded domain

Consider the system $$\left\{ \begin{gathered} u_t - \Delta u = v^p , in Q = \{ (t, x), t > 0, x \in \Omega \} , \hfill \\ v_t - \Delta v = u^q , in Q , \hfill \\ u(0, x) = u_0 (x) v(0, x) = v_0 (x) in \Omega , \hfill \\ u(t, x) = v(t, x) = 0 , when t \geqslant 0, x \in \partial \Omega , \hfill \\ \end{gathered} \right.$$ (S) where Ω is a bounded open domain in ℝN with smooth boundary, p and q are positive parameters, and functions u0 (x), v0(x) are continuous, nonnegative and bounded. It is easy to show that (S) has a nonnegative classical solution defined in some cylinder QT=(0, T) × Ω with T ⩽ ∞. We prove here that solutions are actually unique if pq ⩾ 1, or if one of the initial functions u0, v0 is different from zero when 0 1, solutions may be global or blow up in finite time, according to the size of the initial value (u0, v0).

Any large solution of a non-linear heat conduction equation becomes monotonic in time

SynopsisIn this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) &gt;0, Q(u) &gt; 0 for u &gt; 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.