Large time behavior of solutions to the nonlinear heat equation with absorption with highly singular antisymmetric initial values

In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $ u_t - \Delta u + |u|^\alpha u =0$, where $u=u(t,x)\in {\mathbb R}, $ $(t,x)\in (0,\infty)\times{\mathbb R}^N$ and $\alpha>0$. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables $x_1,\; x_2,\; \cdots,\; x_m$ for some $m\in \{1,2, \cdots, N\}$, such as $u_0 = (-1)^m\partial_1\partial_2 \cdots \partial_m|\cdot|^{-\gamma} \in {{\mathcal S'}({\mathbb R}^N)}$, $0<\gamma0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $\Omega_m = \{x \in {{\mathbb R}^N} : x_1, \cdots, x_m>0\}$, and then to extend the results by reflection to solutions on ${{\mathbb R}^N}$ which are antisymmetric. We show that the large time behavior depends on the relationship between $\alpha$ and $2/(\gamma+m)$, and we consider all three cases, $\alpha$ equal to, greater than, and less than $2/(\gamma+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.


Introduction
In this paper we study the long time behavior of solutions to the nonlinear heat equation with absorption, where u = u(t, x) ∈ R, (t, x) ∈ (0, ∞) × R N and α > 0, which are antisymmetric with respect to the variables x 1 , x 2 , • • • , x m for some m ∈ {1, 2, • • • , N }.Our goal is to see how some well-known results [1,4,5,6] for the long time behavior of solutions to (1.1) carry over with the additional hypothesis of antisymmetry.For example, some of the results in the cited works concern positive solutions.We will see that these results have analogues for antisymmetric solutions which are positive on an appropriate sector in R N .In particular, these solutions are not positive on R N .Moreover, in many cases the range of allowable powers α > 0 will be larger with the additional hypothesis of antisymmetry than without.Also, the condition of antisymmetry allows consideration of a class of highly singular initial values.Our previous paper [9] considered the linear heat equation on R N with antisymmetric solutions.The results and the theoretical framework from [9] were applied to the nonlinear heat equation with source term u t − ∆u − |u| α u = 0, (1.2) in [12].In the current paper, these ideas are applied to (1.1).We mention that this approach was earlier developed in [11] where solutions to (1.2) with antisymmetric initial values of the form u 0 = (−1) m ∂ 1 ∂ 2 • • • ∂ m δ were studied.In the current paper, as in [9,12], initial values of the form , for some 0 < γ < N , are considered.
In order to state our results precisely, we begin by recalling the definition of an antisymmetric function.
where T i , i ∈ {1, 2, • • • , N }, denote the operator We denote the set of functions antisymmetric with respect to x 1 , • • • , x m by A function on R N which is antisymmetric with respect to x 1 , x 2 , • • • , x m , for some m ∈ {1, 2, • • • , N }, is determined by its values on Ω m , the sector of R N defined by (1.5) Note that by definition, an antisymmetric function must take the value 0 on the boundary ∂Ω m .
Since the operators T i defined above commute with the operations in equation (1.1), the study of antisymmetric solutions to (1.1) reduces to the study of solutions on Ω m with Dirichlet boundary conditions.This point is discussed in detail in Section 3 of [12], and that discussion applies as well to the heat equation with absorption.Moreover, as in [12], we will construct certain classes of antisymmetric solutions to (1.1) on R N by constructing solutions on Ω m and extending them to R N by antisymmetry.
Since both the present paper and [12] are based on the framework developed in [9], we need to recall some definitions and notation used in [9].Let ρ m be the weight function defined on Ω m by where 0 < γ < N .We consider the Banach space endowed with the norm ψ Xm,γ = ρ m ψ L ∞ (Ωm) , for all ψ ∈ X m,γ .The closed ball of radius M on X m,γ is denoted by B m,γ,M = {ψ ∈ X m,γ such that ψ Xm,γ ≤ M }. (1.7) As observed in [9, p. 344], B ⋆ m,γ,M the closed ball B m,γ,M endowed with the weak ⋆ topology of X m,γ , is a compact metric space (hence complete and separable).
The function ψ 0 defined on Ω m by where c m,γ = γ(γ +2) • • • (γ +2m−2), will play a central role.It is homogeneous of degree −(γ +m), belongs to X m,γ and satisfies ψ 0 Xm,γ = c m,γ .Moreover, for all σ, λ > 0, and so D σ λ ψ 0 Xm,γ = λ σ−(γ+m) c m,γ .Its interest lies in the fact that (1.12) The heat semigroup on Ω m , denoted e t∆m , is given by e t∆m ψ(x) = Ωm K t (x, y)ψ(y)dy , (1.13) for all t > 0, where for all λ > 0 and σ > 0, and for future use we note the following identity, which is immediate to verify, for all t > 0 and all x ∈ Ω m .In terms of behavior on the sectors Ω m , our goal is to study the well-posedness of the equation (1.1) on the space X m,γ and to obtain results on the large time behavior of solutions in the three cases α = 2/(γ + m), α > 2/(γ + m) and α < 2/(γ + m).By interpreting these results for antisymmetric solutions on R N , we will extend some know results, [4, Theorem 1.3, Theorem 1.4] and [6], in the case m = 0. We now describe these results in detail.
In Section 2, we consider the Cauchy problem (1.17) It is well known that, given any which is a classical solution of (1.1) on R N for t > 0 and such that u(0) = u 0 , which we denote by where u(t) = u(t, •).Likewise, for any u 0 ∈ C 0 (Ω m ), there exists a unique function which is a classical solution of (1.1) for t > 0 and such that u(0) = u 0 .This defines a global semiflow S m (t) on C 0 (Ω m ).In other words, where u(t) = u(t, •) is the solution of (1.1) with initial value u 0 ∈ C 0 (Ω m ).In fact, existence and uniqueness of solutions in C 0 (Ω m ) follows from the existence and uniqueness of solutions in u 0 ∈ C 0 (R N ) since S(t) preserves antisymmetry: it suffices to consider the anti-symmetric extension of u 0 ∈ C 0 (Ω m ) to an element of C 0 (R N ) ∩ A.
Similarly, given any u 0 ∈ L q (Ω m ), 1 ≤ q < ∞, we deduce by Kato's parabolic inequality (see Lemma 8.1 and Corollary 8.2 in the appendix) and the fact that D(Ω m ) is dense in L q (Ω m ), that there exists a unique u ∈ C([0, ∞), L q (Ω m )) which is a classical solution of (1.1) for t > 0 and such that u(0) = u 0 .Alternatively, see [6, Proposition 1.1, p. 261] for a proof using accretive operators.Again by preservation of antisymmetry, the result of [6], valid for R N , holds also on Ω m .Thus, the semi-flow S m (t) extends to L q (Ω m ) and formula (1.19) is valid also for u 0 ∈ L q (Ω m ).
Here we consider initial data u 0 ∈ X m,γ .Our first main result is the following.
In addition, the following properties hold.
where v is the solution of (1.1) with initial value v 0 satisfying (i) and (ii).
(iv) There exists u 0 Xm,γ for all t > 0 and for all u 0 ∈ X m,γ .(v) The solution u(t) satisfies the integral equation for all t > 0, where the integrand is in , where v is the solution of (1.1) with initial value v 0 satisfying (i) and (ii).
In other words, the nonlinear operators S m (t), t > 0, extend in a natural way to X m,γ .We remark that in the case α < 2/(γ + m), this well-posedness result was established in [12,Theorems 2.3 and 2.6] by a different method and with plus and minus sign in the term of the nonlinearity.Furthermore, the analogous results on the whole space R N follows from [1,Theorem 8.8,p. 536].
We also establish the continuous dependence properties of solutions of equation (1.1) with initial values in X m,γ .
, for all t > 0, where B ⋆ m,γ,M denotes the compact metric space topology induced by the weak* topology on B m,γ,M .
It is well-known that any solution u(t) of (1.1), for example as constructed in Theorem 1.2, is always bounded by the spatially independent solution, more precisely for all t > 0, throughout the spatial domain of existence.See for example [6, page 261].In addition, it is clear from Theorem 1.2 that if u is the solution of (1.1) with positive initial data u 0 ≥ 0 then for any t > 0. We have the following upper estimate for solutions of (1.1) which combines (1.22) and (1.21) into one estimate which implies them both.Its proof is given in Section 3.
Then the solution u of (1.1) with initial data u(0) = u 0 satisfies the following upper estimate for all t > 0, and all x ∈ Ω m .
After proving global well-posedness of the Cauchy problem (1.17), i.e.Theorems 1.2 and 1.4, we seek to describe the large time behavior of solutions of (1.1) on Ω m with initial values in X m,γ .Our basic approach is to study the effect of certain space-time dilations on such a solution, and to relate the resulting behavior to the effect of related spatial dilations on the initial value.In particular we consider the space-time dilation operators Γ σ λ , λ > 0, defined by for all λ, σ > 0. If u ∈ C((0, ∞), C 0 (Ω m )) is solution of the equation (1.1) then Γ σ λ u is solution of (1.1) if and only if σ = 2/α.Moreover, if a solution u has initial value u 0 , either in the sense of C 0 (Ω m ) or in some more general sense, then Γ , for all λ > 0, and the uniqueness of solutions of (1.1) implies that Γ 2/α λ u coincides with S m (•)D 2/α λ u 0 .Thus, we have the following relation for all u 0 ∈ X m,γ .We emphasize that at this point there is no assumed relationship between α and m.Formula (1.25) holds for any semiflow generated by (1.1) in place of S m (•), as long as the space of initial values is invariant under the dilations D 2/α λ and initial values give rise to unique solutions.A solution u of (1.1) is self-similar if Γ 2/α λ u = u, for all λ > 0, or equivalently if where f (x) = u(1, x) is called the profile of u.It follows that if a self-similar solution u of (1.1) has initial value u 0 , then D 2/α λ u 0 = u 0 , for all λ > 0, i.e. u 0 is homogeneous of degree −2/α.Conversely, if u 0 is homogeneous of degree −2/α and u(t) is a solution with initial value u 0 in some appropriate sense, then Γ 2/α λ u has the same initial value, for all λ > 0. Assuming that uniqueness of solutions having a given initial value has been proved in the appropriate class of functions, one then concludes that u = Γ 2/α λ u, for all λ > 0, i.e. that u is a self-similar solution.More generally, we say that a solution u of (1.1) is asymptotically self-similar if in some appropriate sense, and that U is also a solution to (1.1).If so, the limit is necessarily a self-similar solution.See Section 3 of [5] for a discussion of several equivalent definitions of asymptotically self-similar solutions.Formally, if we put t = 0 in (1.27), we obtain that where ϕ = U (0) is homogeneous of degree −2/α.In the Section 4 we study the long time asymptotic behavior of solutions to (1.1) with initial values in X m,γ in the case α = 2/(γ + m).The first result shows that (1.28) implies (1.27).
Suppose that there exists ϕ ∈ B m,γ,M such that lim λ→∞ It follows that ϕ is homogeneous of degree −(γ + m) and that the solution u(t) = S m (t)ψ is asymptotically self-similar to the self-similar solution U (t) = S m (t)ϕ.
As is by now well established [3,4,5], the notion of asymptotically self-similar solution can be naturally extended by allowing different limits in (1.28) and (1.27) along different sequences (λ n ) n≥0 , with λ n → ∞.The next step in our analysis it to generalize Theorem 1.6 in this fashion.To accomplish this, for u 0 ∈ X m,γ and M ≥ u 0 Xm,γ , we consider the set of all accumulation points of D γ+m λ u 0 , as λ → ∞, given by Since B ⋆ m,γ,M is a compact metric space, Z γ (u 0 ) is nonempty compact subset, for all u 0 ∈ X m,γ , and independent of M ≥ u 0 Xm,γ by [9, Proposition 3.1, p. 356].In particular, if u 0 is homogeneous of degree −(γ + m), then Z γ (u 0 ) = {u 0 }.We set u(t) = S m (t)u 0 and we also define the omega-limit set of all accumulation points of Γ γ+m The relation (1.25) and Theorem 1.4 are the essential elements needed to investigate the relationship between Q γ (u 0 ) and Z γ (u 0 ), which is given by our next main result.
The last relation shows that in the case α = 2/(γ + m) the complexity in the large time behavior of a solution, as expressed in Q γ (u 0 ), is determined by the complexity in the spatial asymptotic behavior of its initial value as expressed in Z γ (u 0 ).Furthermore, Theorem 1.7 above is inspired from [4, Theorem 1.3, p. 83] which requires α ≥ 2/N , and we observe that in Theorem 1.7, if γ + m > N , then α < 2/N .Since B ⋆ m,γ,M is separable and Z γ (u 0 ) can contain any countable subset of B ⋆ m,γ,M , we show that Z γ (U 0 ) = B ⋆ m,γ,M for some choice of U 0 ∈ B ⋆ m,γ,M .Using [9, Theorem 1.4, p. 345], we obtain the following result.
It is clear from (1.31) that for u 0 ∈ X m,γ , the transformations most likely to yield some nontrivial asymptotic behavior are Γ γ+m λ .In other words, we still need to study Q γ (u 0 ) as given by (1.30), and likewise Z γ (u 0 ) as given by (1.29).However, we cannot expect the relationship between these two objects to be given as in Theorem 1.7 since the transformations do not preserve solutions of (1.1).
If u is a solution of (1.1) then v = Γ γ+m λ u is the solution of the equation , it follows that as λ → ∞, the function v satisfies an equation which approaches the linear heat equation.Hence, we should not be surprised if in this case Q γ (u 0 ) and Z γ (u 0 ) are related by the linear heat equation.The next theorem makes this idea precise, both in the asymptotically self-similar case, and the more general case of arbitrary We then have the following conclusions.
In Section 6 of this paper, we consider the case α < 2/(γ + m).As in the case of Theorem 1.10, the transformations which leave solutions invariant, i.e.Γ 2/α λ , do not leave invariant the norm of X m,γ , which is the space where the solution lives.Nonetheless, unlike in the case α > 2/(γ + m), the transformations Γ 2/α λ reveal nontrivial asymptotic behavior.Because of (1.31), to study this asymptotic behavior, we need to leave the context of the space X m,γ .This is best illustrated by the result of Gmira and Véron [6] in the case of R N .If we express the upper bound (1.21) in terms more suggestive of the long-time asymptotic behavior of the solution, we see that, considering only positive solutions, . (1.34) The main result of [6] can be stated as follows.Suppose α < 2 N .Let u 0 ∈ L q (R N ) for some 1 ≤ q < ∞, or C 0 (R N ), with u 0 ≥ 0, be such that for every k > 0, there exists R 0 > 0 such that is the resulting solution of (1.1), then uniformly on compact subsets of R N .In light of the upperbound (1.34), the result (1.36) is rather sharp.
In the case of the sector Ω m , we have the following result, where C b,u 0 (Ω m ) denotes the space of bounded uniformly continuous functions on Ω m which are zero on ∂Ω m .
Let u 0 ∈ X m,γ with u 0 ≥ 0, and let u(t) = S m (t)u 0 be the resulting solution of (1.1) as given by Theorem 1.2.Suppose that there exist R 0 > 0 and c 0 > 0 such that where ψ 0 is given by (1.10).Then uniformly on compact subsets of Ω m , where g ∈ C b,u 0 (Ω m ) is the profile of the self-similar solution of (1.1) given by Proposition 6.3.
Remark 1.12.The condition (1.38) implies that, for any c > 0, Remark 1.13.Using (6.15) below and (1.14), we have that g in (1.39) satisfies the explicit bound for all 0 < ε < 1, where In the Section 7 of this paper, we reinterpret the results of the previous sections on the global well-posedness and the asymptotic behavior of S m (t)u 0 , u 0 ∈ X m,γ , in the case of antisymmetric functions defined on the whole space R N .Recall that the heat semigroup on R N is given by for all t > 0 and x ∈ R N .The heat semigroup e t∆ was studied in [4] on the space with 0 < σ < N .It was observed in [9], that we can consider the case N ≤ σ < 2N for some class of antisymmetric initial values in W σ .See [9, Corollary 1.7, p. 346] and the discussion just after.If ψ : Ω m → R, we denote by ψ its pointwise extension to R N which is antisymmetric with respect to , ψ has a natural interpretation as an element of S ′ (R N ).See [9,Definition 1.6,p. 346].We also define the space with the norm ϕ Xm,γ = ϕ | Ωm Xm,γ , for all ϕ ∈ X m,γ .We also consider, We denote by B ⋆ m,γ,M the ball B m,γ,M endowed with the weak ⋆ topology.B ⋆ m,γ,M inherits the metric space structure from B ⋆ m,γ,M .In addition, we observe that X m,γ ⊂ W γ+m with continuous injection.However the two norms are not equivalent.On the other hand, B ⋆ m,γ,M ⊂ (B γ+m M ) ⋆ where (B γ+m M ) ⋆ denote the closed ball of radius M on W γ+m endowed with the weak ⋆ topology, but here the metric on B ⋆ m,γ,M is equivalent to the one it inherits from the metric space (B γ+m M ) ⋆ .See Proposition 7.1 below.
The heat semigroup e t∆ is well-defined on X m,γ and e t∆m ψ = e t∆ ψ.
In addition, the following properties hold.
(iii) For all w 0 ∈ X m,γ , |v(t) − w(t)| ≤ e t∆ |v 0 − w 0 | ; where w is the solution of (1.1) with initial value w 0 satisfying (i) and (ii).(iv) v(t) satisfies the integral equation Since X m,γ ⊂ W γ+m , where W γ+m is given by (1.42), the last result gives a new class of initial values for which we have global well-posedness of solutions in the case α < 2/N (when γ + m > N ).See [1] and [4, Section 4] for information about non-uniqueness of solutions in the case α < 2/N .
The semiflow S(t) defined by (1.18) extends to X m,γ as the following.
From the construction in Theorem 1.14 and the uniqueness part we have the following formula for all t > 0 and u 0 ∈ X m,γ .As in the case of the sectors Ω m , i.e. the flow S(t) depends continuously on the initial values.The following is an adaptation of Theorem 1.4.
We now consider the long-time asymptotic behavior of the solutions described in Theorem 1.14.In analogy with (1.29) and (1.30) above, and using a notation consistent with formulas (1.17) and (1.18) in [3] and [4, Definition 1.2], we make the following definitions.For v 0 ∈ X m,γ we define the ω-limit set of possible asymptotic forms of v 0 , by and the ω-limit set of all limits of Γ γ+m √ t S(1)v 0 , as t → ∞, by The following three theorems are reformulations of the results above on the asymptotic behavior of solutions, adapted from the case of the sectors Ω m to the case of antisymmetric functions on R N , in the three cases: α equals, is greater than, and is less than 2 γ+m .
It follows that It follows that , then ϕ is homogeneous of degree −(γ + m) and the solution v(t) = S(t)v 0 of (1.1) is asymptotic to the self-similar solution of the linear heat equation U (t) = e t∆ ϕ; Let v 0 ∈ X m,γ with v 0 | Ωm ≥ 0, and let v(t) = S(t)v 0 be the resulting solution of (1.1) as given by Definition 1.15.Suppose that there exist R 0 > 0 and c 0 > 0 such that where ψ 0 is given by (1.10).Then uniformly on compact subsets of R N , where g ∈ C b,u (R N ) is the antisymmetric (bounded, uniformly continuous) profile of the self-similar solution of (1.1) given by Proposition 7.3.
Finally, in the appendix, for completeness we give a proof of Kato's parabolic inequality and the main application for which we use it.Also, we present some results which we found during the course of research for this article, which we feel have some independent interest, but which ultimately were not needed for the proofs of the main results.One of them concerns the lowest eigenvalue and corresponding eigenfunction for −∆ on B 1 = {x ∈ Ω m : |x| < 1} with Dirichlet boundary conditions.
The authors wish to thank Philippe Souplet for several very helpful remarks concerning this research.

Existence and continuity properties of solutions
The purpose of this section is to study well-posedness of the equation (1.1) with initial values in X m,γ and to give the proofs of Theorem 1.2 and Theorem 1.4.For this purpose, we need several results from [9], sometimes in a slightly stronger version.The first result below is a slight improvement of [9, Proposition 2.5, p. 353].
In particular, the convergence is also in We write Using the inequality e for all i ∈ {1, • • • , m}, we deduce from (1.14) that for all x, y ∈ Ω m , Therefore, This implies that e t∆m [ηψ] → 0, a.e.pointwise on K, as t → 0.Moreover, by Proposition [9, Theorem 1.1 (i), p. 343], we have Thus, by the dominated convergence theorem, e t∆m [ηψ] → 0 on L 1 (K), as t → 0. On the other hand, since (1 1), we obtain that e t∆m ψ → ψ in L 1 loc (Ω m ), as t → 0. This completes the proof.
We also need to use a stronger version of [9, Lemma 2.6, p. 355], as follows.
The following is a version of [1, Corollary 8.3, p. 531] adapted from R N to Ω m .
for all t > 0 and all x ∈ Ω m .
For the general case, we proceed by scaling and observe that Using formula (1.15) and the inequality (2.5), we obtain This completes the proof.
We will also use the following lemma, which gives a property of convergence in We now give the proof of Theorem 1.2.
Proof of Theorem 1.2.Let u 0 ∈ X m,γ and let (K n ) n≥1 be the sequence of nondecreasing compacts in Ω m defined by: We consider the function where ξ n is a cut-off function satisfying Note that, u 0,n ∈ X m,γ , for all n ≥ 1, and • for a fixed compact K on Ω m , then there exist n 0 such that u 0,n = u 0 on K, for all n ≥ n 0 .
It remain now to show that u(t) → u 0 on L 1 loc (Ω m ), as t → 0. We fix a compact subset K ⊂ Ω m and n such that u 0,n = u 0 on K. Thus, By letting ℓ → ∞ in (2.6), we have that and so K |u(t) − u n (t)| → 0, as t → 0. Since u n ∈ C([0, ∞), L p (Ω m )), we have u n (t) → u 0,n on L p (Ω m ), as t → 0, so that This proves that u(t) converges to u 0 on L 1 loc (Ω m ), as t → 0, and so (i) is proved.Uniqueness: Let s > 0 and u, v two solutions of (1.1) satisfying (i) and (ii).We have that m,γ,M for all s > 0 and u(s), v(s) → u 0 in L 1 loc (Ω m ), as s → 0, it follows from the Lemma 2.4 that the right hand side of the last inequality tends to 0 in C 0 (Ω m ), as s → 0. This gives that |u(t as s → 0, for every fixed t > 0. By uniqueness of the limit, we have u(t) = v(t), for all t > 0.
Additional properties: We next give the proof of the statements (iii), (iv) and (vi).In fact, by (2.8), we have |u(t)| ≤ e t∆m |u 0 |, and so, from Lemma 2.2, we obtain for all t > 0 and x ∈ Ω m .In addition, if u 0 , v 0 ∈ X m,γ , we denote u(t) and v(t) the corresponding solutions.For all n ≥ 1, we let u 0,n = u 0 ξ n and v 0,n = v 0 ξ n where ξ n is the cut-off function defined by (2).Then, for all n ≥ 1, Letting n → ∞ and using Lemma 2.4, we deduce that Finally, assertion (vi) is true since, under the same conditions, |u n (t)| ≤ v n (t), by well-known comparison results.
The following lemma is needed to establish Theorem 1.4.

An upper bound on solutions
In this section we prove Proposition 1.5.This proposition is stated for solutions on the domain Ω m , but in fact is valid for solutions of (1.1), or rather the associated integral equation, on any domain Ω.Accordingly, we state here the more general version.Both the statement and proof are inspired by the statement and proof of [13,Theorem 1].Moreover, we introduce some notation which will be used solely in this section.
Let Ω ⊂ R N be a domain, not necessarily bounded, and let C 0 (Ω) be the space of continuous functions f : Ω → R such that f ≡ 0 on the boundary ∂Ω and f (x) → 0 as |x| → ∞ in Ω.Let e t∆ be the heat semigroup on C 0 (Ω), given by a kernel k t = k Ω t as follows; In particular, if f ∈ L 1 loc (Ω), f ≥ 0, then e t∆ f is likewise defined by formula (3.1).
Proof.Fix 0 < τ < T , and set Since for all x ∈ Ω the measure k τ −t (x, y)dy on Ω, has total mass less than or equal to 1, Jensen's inequality implies that Integrating this last differential inequality on [0, t] we obtain which is the same as This is true for 0 < τ < T and 0 ≤ t ≤ τ .The result follows by setting t = τ > 0.
Remark 3.2.Using an argument similar to the above, one can obtain an analogous estimate for positive solutions of the more general equation where f is a positive, convex, increasing where F −1 is the inverse function of F.

Self-similar asymptotic behavior on sectors
In this section we consider equation (1.1) in the case 2/α = γ + m.Let u 0 ∈ X m,γ and set u(t) = S m (t)u 0 .Using (1.25) we can re-write the definition (1.30) of the ω-limits set Q γ (u 0 ) in the following equivalent form, We begin by proving the Theorem 1.6 which corresponds to the particular case when Q γ (u 0 ) contains one nontrivial element.
Proof of Theorem 1.6.Using limits in the sense of D ′ (Ω m ), we have in C([τ, t]; C 0 (Ω m ), for all 0 < τ < t, so that S m (t)ψ is asymptotically self-similar to the self-similar solution U (t).
We give now the proof of Theorem 1.7.
Proof of Theorem 1.7.Let u 0 ∈ X m,γ and M > 0 be such that . This proves the result.

Linear asymptotic behavior on sectors
In this section, we study the long-time asymptotic behavior of solutions to (1.1) in the case 2/α < γ + m.The key point is that under the dilations D γ+m √ t , which preserve the norm of X m,γ , the integral term in (1.1) decays faster than the difference between the two other terms.This is the content of the next proposition.
Proof.We know that, for all t > 0, Therefore, using (1.15), we have On the other hand, estimating as in (2.10), we see that By Corollary 2.3, we deduce that

It follows that
Since α(γ + m) > 2, we see that This proves the result.

Nonlinear asymptotic behavior on sectors
In this section we consider the equation (1.1) with non-negative initial value u 0 ∈ X m,γ in the case α < 2/(γ + m), and our goal is to prove Theorem 1.11.First however, we need to show that the hypothesis on the initial condition u 0 , which gives a lower bound for large |x|, implies a lower bound on the resulting solution at any fixed positive time.The key point is the behavior near the boundary.We prove the following result.Proposition 6.1.Let u 0 ∈ X m,γ , with u 0 ≥ 0, and suppose that there exist ρ > 0 and c > 0 such that for all x ∈ Ω m with |x| ≥ ρ, where ψ 0 is given by (1.10).Let u(t, •) = u(t) = S m (t)u 0 be the resulting solution of (1.1) as given by Theorem 1.2, and fix any t 0 > 0. It follow that v 0 ≡ S m (t 0 )u 0 verifies the condition for some c ′ > 0, where the constant c ′ may depend on t 0 .
We refer the reader to [8] for results of this type on a general domain.The present situation differs from that in [8] in that the sector Ω m does not have the required regularity, and also that here we include the possibility that u 0 could be identically zero on a bounded subset of Ω m .Unlike [8], our proof makes use of the explicit form of the kernel for the heat semigroup on Ω m .
Proof.We first note that it suffices by comparison to prove this for the specific initial value where ψ 0 is given by (1.10), and ρ > 0 is arbitrary.To accomplish this, we first prove that for any fixed t 0 > 0, v 0 = e t 0 ∆m u 0 verifies (6.2),where u 0 is given by (6.3).For this purpose, since the estimate is linear, the value of c > 0 in (6.3) is of no importance.Thus, we consider e t∆m u 0 on Ω m given by (1.13) and (1.14), where u 0 is given by (6.3).Using the fact that e s − e −s ≥ 2s for all s ≥ 0, we see that if x, y ∈ Ω m and 1 ≤ i ≤ m (so that x i ≥ 0 and y i ≥ 0), then It follows that u 0 (y)dy This shows in particular that for any t > 0, e t∆m u 0 satisfies (6.2), but only on any give bounded set in Ω m .
We turn our attention to the case where |x| is large.
Since this calculation is for large |x| we may suppose that and so we see that Also, we want to use the specific formula in (6.3), so we impose where ρ is as in (6.3).Hence We can now calculate.
We first need to examine the integral y i dy i for 1 ≤ i ≤ m, under two different circumstances, 0 < x i < 1 and x i ≥ 1.Consider first the case x i ≥ 1.We have, since We next consider the case x i ≤ 1.We have, by (6.4), We second need to examine the integral We have, if in addition We In all cases, we have It therefore follows from (6.11), (6.12), (6.13) that, if x ∈ Ω m , then Combining (6.5) and (6.14), we obtain that for any fixed t > 0, e t∆m u 0 satisfies (6.2).We next show the same result for u(t, •) = u(t) = S m (t)u 0 be the resulting solution of (1.1), where u 0 is given by (6.3).To do so, set w(t) = e µt u(t), where µ = [cρ γ+m ] α ≥ u 0 α L ∞ (Ωm) .Since u(t) ≤ u 0 L ∞ (Ωm) for all t > 0, we have u(t) α ≤ µ for all t > 0. It follows that w ′ (t) = e µt u ′ (t) + e µt µu(t) ≥ e µt u ′ (t) + e µt u(t) α u(t) = e µt ∆u(t) = ∆w(t).
Hence w(t) ≥ e t∆m w(0) = e t∆m u 0 .In other words u(t) ≥ e −µt e t∆m u 0 , which implies the desired result.Remark 6.2.In addition to being well-posed in C 0 (Ω m ), in L q (Ω m ) for 1 ≤ q < ∞, as noted in the introduction, and in X m,γ , as per Theorem 1.2, equation (1.1) is globally well-posed in L ∞ (Ω m ) in the following sense.For every u 0 ∈ L ∞ (Ω m ), there is a unique solution u ∈ C((0, ∞); C b,u 0 (Ω m )) of the integral equation (1.20), where C b,u 0 (Ω m ) denotes the closed subspace of L ∞ (Ω m ) of bounded, uniformly continuous functions on Ω m which are zero on ∂Ω m , but not necessarily as |x| → ∞.This solution has the following additional properties: the function u is a classical solution of (1.1) on (0, ∞) × Ω m , u(t) − e t∆m u 0 L ∞ (Ωm) → 0 as t → 0, and |u(t)| ≤ (αt) 1/α , for all t > 0. One way to see this is first to establish the corresponding result on L ∞ (R N ), but of course with C b,u (R N ) instead of C b,u 0 (Ω m ), and then to restrict to anti-symmetric functions on R N .The result on R N follows from standard arguments, i.e. contraction mapping, parabolic regularity, and comparison.We refer the reader to Appendices B and C of [2] for detailed information about e t∆ on C b,u (R N ).In particular, [2,Lemma B.1] establishes that e t∆ h ∈ C b,u (R N ) for all h ∈ L ∞ (R N ) and [2, Theorem C.1], which still valid for the nonlinear heat equation with absorption, establishes the necessary regularity.Proposition 6.3.Let m ∈ {1, 2, • • • , N } and α > 0. There exists a self-similar solution V (t, x) = t −1/α g( x √ t ) of equation (1.1) such that g ∈ C b,u 0 (Ω m ), the space of bounded uniformly continuous functions on Ω m which are zero on ∂Ω m , g ≥ 0, and α −1/α e ∆m h ≤ g ≤ (αǫ) −1/α e (1−ǫ)∆m h (6.15) for all 0 < ǫ < 1, where h(x) = 1 is the constant function on Ω m .The self-similar solution V is characterized by where v is the solution to (1.1) with initial value v 0 = h, as described in Remark 6.2, the dilations Γ 2/α λ are defined by (1.24), and where the limit (6.16) is uniform on compact subsets of (0, ∞)×Ω m .
We observe that in the case m = 0, the corresponding self-similar solution is (αt) −1/α .
Proof of Theorem 1.11.By the hypotheses on u 0 and by Proposition 6.1, we have that for t 0 > 0, on Ω m and we know that u(t 0 ) ∈ C 0 (Ω m ).Up to a translation in time and since we are concerned with the large time behavior, we may suppose that u 0 ∈ X m,γ ∩ C 0 (Ω m ), u 0 ≥ 0 and verifies (6.29).
In fact, it suffices to assume , and that u(t, x), v(t, x) and w(t, x) are the solutions of (1.1) with initial values respectively u 0 , v 0 and w 0 ≡ c ′ .We know by comparison that uniformly on compact subsets of Ω m , then clearly, since by Proposition 6.3 also uniformly on compact subsets of Ω m , it follows that uniformly on compact subsets of Ω m .Thus, we now assume the initial value u 0 ∈ X m,γ is given by (6.30), and we denote by u(t) = S m (t)u 0 be the resulting solution of (1.1) given by Theorem 1.2.We use a method introduced in [7].Consider the space-time dilations functions defined by (1.24) with σ = 2/α: In particular, u λ is the solution of (1.1) with initial data ) is an increasing function in λ > 0, for all x ∈ Ω m .(It's the minimum of two functions which are obviously increasing in λ.) Consequently, the solutions u λ (t, x) are likewise increasing in λ > 0. We note also that the solutions w λ (t, x) are increasing in λ > 0 (as in the proof of Proposition 6.3), where w is the solution with initial value w 0 ≡ c ′ as above. Since where V is the self-similar solution of Proposition 6.3, it follows that the following limit exists and for all λ > 0.Moreover, by parabolic regularity and standard compactness arguments, the limit (6.34) is uniform on compact subsets of (0, ∞) × Ω m .
We next wish to show that For this we need to obtain a lower bound for U.
Let A > 0, and consider the family of truncated initial values Let z A λ be the solution of (1.1) with initial data u A 0,λ .By comparison principle and (6.35) for every λ > 0 and A > 0.Moreover, it is clear from (6.33) that for each fixed A > 0, the initial values u A 0,λ (x) are an increasing function of λ > 0, and so therefore must be the solutions z A λ (t, x).Furthermore, the initial values satisfy the monotone limit lim λ→∞ u A 0,λ (x) = A, (6.38) and the corresponding solutions converge in a monotone fashion to some function lim λ→∞ z A λ (t, x) = Z A (t, x) ≤ U (t, x).(6.39) We next consider the integral equation satisfied by z A λ (t), i.e. equation (1.20) with initial value u A 0,λ .Using (6.38) and (6.39) along with the monotone convergence theorem, we see that Z A (t) satisfies which implies, along with (6.39), that V (t, x) ≤ U (t, x).Thus by (6.34), V (t, x) = U (t, x).
We have the following result, which is an application of Kato's inequality.Finally, we give two results which we found during our research for this article, and which we believe have an independent interest, but which ultimately were not needed for the proofs of the main results.
Consider the eigenvalue problem, on some domain B ⊂ R N − ∆H = ΛH (8.3) where Λ ∈ R. We look for a solution of the form where r = (x 2 1 + x 2 2 + • • • + x 2 N ) 1/2 .We note that for 1 ≤ i ≤ m, where xi means that x i is missing from the product, and and if m < i ≤ N , then Proof.By (1.10) the function ψ 0 can be written in the form (8.4), that is The result follows then by (8.5).

e
− |x i −y i | 2 4t − e − |x i +y i | 2 4t .(1.14) See, for example, [11, Proposition 3.1, p. 514].It is well-known that e t∆m is a C 0 semigroup on C 0 (Ω m ), the space of continuous functions f : Ω m → R such that f ≡ 0 on the boundary ∂Ω m and f (x) → 0 as |x| → ∞ in Ω m .It is also well defined on X m,γ and e t∆m : X m,γ → C 0 (Ω m ) ∩ X m,γ is continuous, for all t > 0. See [9, Theorem 1.1, p. 343].We recall the commutation relation between e t∆m and the operators D σ λ , D σ λ e λ 2 t∆m = e t∆m D σ λ (1.15) (1.45)    See[9, Proposition 5.1, p. 361].The last formula is the key to the study the equation (1.1) in the space X m,γ .The following result is essentially a reformulation of Theorem 1.2 for antisymmetric functions on R N .Theorem 1.14

Z
A (t) = e t∆m A − t 0 e (t−s)∆m (|Z A (s)| α Z A (s)) ds,(6.40)i.e.Z A is the solution of (1.20) with initial value Z A (0) ≡ A on Ω m .We know by (the proof of) Proposition 6.3 that lim A→∞ Z A (t) = V (t),
next consider the case |x i | ≤ 1.By the inequality, e −