Large time behavior of ODE type solutions to nonlinear diffusion equations

Consider the Cauchy problem for a nonlinear diffusion equation \begin{equation} \tag{P} \left\{ \begin{array}{ll} \partial_t u=\Delta u^m+u^\alpha&\quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0)=\lambda+\varphi(x)>0&\quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} where $m>0$, $\alpha\in(-\infty,1)$, $\lambda>0$ and $\varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N)$ with $1\le r<\infty$ and $\inf_{x\in{\bf R}^N}\varphi(x)>-\lambda$. Then the positive solution to problem (P) behaves like a positive solution to ODE $\zeta'=\zeta^\alpha$ in $(0,\infty)$ and it tends to $+\infty$ as $t\to\infty$. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.


1.
Introduction. Let u be a positive solution to the Cauchy problem for a nonlinear diffusion equation where m > 0, α ∈ (−∞, 1), λ > 0 and ϕ ∈ BC(R N ) ∩ L r (R N ) with 1 ≤ r < ∞ and inf x∈R N ϕ > −λ. For µ > 0, let ζ µ = ζ µ (t) satisfy ζ = ζ α in (0, ∞) and ζ(0) = µ, that is, Then the solution u to problem (P) satisfies lim t→∞ u(x, t) = ∞ uniformly for x ∈ R N and Indeed, the comparison principle implies that as t → ∞ uniformly for x ∈ R N , and (1.2) holds. (See also [1,13,18,21] for related results.) In this paper we obtain the precise description of the large time behavior of the solution and show that the function U = U (x, t) defined by behaves like a rescaled solution to the heat equation as t → ∞ if m ≥ α.
For any 1 ≤ r ≤ ∞, let · r be the usual norm of L r := L r (R N ). For any K ≥ 0, we define L 1 K := {f ∈ L 1 loc (R N ) : |||f ||| K < ∞}, where The large time behavior of ODE type solutions was studied in [11], which deals with the Cauchy problem for the heat equation with absorption [11,Proposition 3.1]). Here η λ is a solution of η = −η γ in (0, ∞) with η(0) = λ. In [11] the second author of this paper and Kobayashi introduced a function and proved the following: as t → ∞, for any q ∈ [1, ∞]. We can improve statement (b) with the aid of [10] and we obtain: as t → ∞, for any q ∈ [1, ∞]. By statements (a), (b) and (b') we see that the function V behaves like a solution of the heat equation as t → ∞. In particular, by statements (b) and (b') we obtain the higher order asymptotic expansion of the function V as t → ∞.
In this paper we say that u is a solution to problem (P) if The main purpose of this paper is to obtain the precise description of the large time behavior of the function U (see (1.4)). There are many results on the large time behavior of the solutions to the porous medium equation ∂ t u = ∆u m (see e.g. [20]), however it still seems difficult to obtain the precise description of the large time behavior of the solutions such as the higher order asymptotic expansions of the solutions. In this paper, for the case of m ≥ α, taking an advantage of the fact that the solution behaves like a solution to ODE ζ = ζ α and developing the arguments in [9,10,11], we show that the function U behaves like a rescaled solution to the heat equation as t → ∞ and obtain the higher order asymptotic expansions of U .
We state the main results of this paper. In Theorem 1.1 we study the large time behavior of U in the case of ϕ ∈ L r (R N ) with r > 1 and we show that U behaves like a rescaled function of e t∆ ϕ as t → ∞ if m ≥ α. Assume ϕ ∈ BC(R N ) ∩ L r (R N ) for some r ∈ (1, ∞). Then as t → ∞, for any q ∈ [r, ∞].
In Theorem 1.2, under the assumption that m ≥ α and ϕ ∈ L 1 K with K ≥ 0, we prove that U behaves like a suitable multiple of a rescaled Gauss kernel as t → ∞ and obtain the higher order asymptotic expansions of U .
In Theorem 1.2 the asymptotic profile of U is described by the Gauss kernel G and its derivatives and it is independent of the Barenblatt solutions even if m = 1.
(ii) For the case of m < α, U converges to a nontrivial function as t → ∞ and it does not necessarily decay as t → ∞. See Remark 5.1.
Although Theorems 1.1 and 1.2 are new even in the case of m = 1, the novelty of this paper is to obtain the precise description of the large time behavior of U in the case of m = 1. For the solution u to (P), we set w(x, τ ) := U (x, t(τ )), where t(τ ) is the inverse function of σ(t) (see (1.7)), that is, (1.12) Here we remark that H(τ, w, ∇w) = 0 if m = 1 and . (1.13) In [9], developing the arguments in [7] and [8], the second author of this paper and Kawakami studied the Cauchy problem to nonlinear diffusion equations of the form and established a method to obtain the higher order expansions of the solution behaving like a suitable multiple of the Gauss kernel as t → ∞. (See also [10] and [12].) However, in the case of m = 1, due to the nonlinear term div H(τ, w, ∇w), we can not apply the arguments in [7]- [10] and [12] to problem (1.11) directly. Indeed, it is difficult to apply their arguments to the Cauchy problem for nonlinear diffusion equations of the form On the other hand, showing that A(τ, w) → 1 as τ → ∞ uniformly on R N , we can apply the parabolic regularity theorems to w and obtain gradient estimates of w. Then we can regard Cauchy problem (1.11) as the Cauchy problem to a nonlinear heat equation of the form . Furthermore, using nice properties of H(τ, w, ∇w) (see e.g. (4.1) and (4.11)), we apply the arguments in [10] and [11] to obtain the precise description of the large time behavior of w. Then we can complete the proofs of Theorems 1.1 and 1.2.
Our arguments are completely different from the arguments well-used in the study of the large time behavior of the solutions to nonlinear diffusion equations related to the porous medium equation. Compare with e.g. [3,4,5,6,14,16,17,19,20].
The rest of this paper is organized as follows. In Section 2 we recall some results on the higher order asymptotic expansion of the solutions to the heat equation with an inhomogeneous term. In Section 3 we obtain some preliminary lemmas on the behavior of w. In Sections 4 and 5 we prove Theorems 1.1 and 1.2, respectively.

2.
Preliminaries. In this section we introduce some notation and recall some properties of the Gauss kernel. Furthermore, we recall some lemmas on the higher order asymptotic expansion of the solutions to the heat equation with an inhomogeneous term.
For any nonnegative functions f and g in (a, ∞), where a ∈ R, we say that f ∼ g as t → ∞ if lim t→∞ f (t)/g(t) = 1. Furthermore, by the letter C we denote generic positive constants and they may have different values also within the same line.
We collect some estimates on the Gauss kernel. By the explicit representation of G (see (1.5)), for any ν ∈ M and j = 0, 1, 2, · · · , we have Then it follows that where 1 ≤ q ≤ ∞ and ≥ 0. Furthermore, the Young inequality together with (2.2) implies the following properties: • Let ν ∈ M and 1 ≤ p ≤ q ≤ ∞. Then there exists a constant c |ν| > 0 independent of p and q such that for ϕ ∈ L p (R N ). In particular, e t∆ ϕ q ≤ ϕ q (1 ≤ q ≤ ∞) holds for t > 0. • Let K ≥ 0 and δ > 0. Then there exists a constant C > 0 such that for any T > 0.
(i) For any ν ∈ M with |ν| ≤ K, there exists a constant C 1 > 0 such that for almost all t > 0.
Let j ∈ {0, 1} and T * > 0. Then there exists a constant C 2 > 0 such that, for any > 0 and T ≥ T * , for all sufficiently large t > T . In particular, if 3. Preliminary estimates of solutions. Assume m ≥ α. Let u be the solution to (P). Due to (1.3), the diffusion coefficient mu m−1 of the nonlinear diffusion equation in (P) is bounded and is not degenerate in R N × (0, T ) for any T > 0. By the parabolic regularity theorems we see that u is smooth in R N × (0, ∞) (see also Lemma 3.3). Let w = w(x, τ ) be as in Section 1, that is, On the other hand, for any µ > 0, it follows from (1.1) that Then, by (1.3) we see that By Lemma 3.1 we apply the Taylor theorem to the function F := F (τ, w) defined by (1.12). Then, for any x ∈ R N and τ > 0, we can find θ ∈ (0, 1) such that  3) and (1.13) we can find C > 0 such that for x ∈ R N and τ > 0. These imply that where C * > 0. Then we have the following decay estimates of w.
Lemma 3.2. Assume the same conditions as in Theorem 1.1. Let w be as in (3.1). (3.11) Proof. Let w ± satisfy where C * and D are as in (3.9). Applying the comparison principle to (1.11), we have .

4.
Proof of Theorem 1.1. We prove Theorem 1.1. By Lemma 3.1 and the Taylor theorem, for any x ∈ R N and τ > 0, we can find θ ∈ (0, 1) such that Then, by Lemma 3.3 and (3.6) we have  for τ > 0.