Global rough solution for L 2 -critical semilinear heat equation in the negative Sobolev space
aa r X i v : . [ m a t h . A P ] M a r GLOBAL ROUGH SOLUTION FOR L -CRITICAL SEMILINEAR HEATEQUATION IN THE NEGATIVE SOBOLEV SPACE AVY SOFFER, YIFEI WU, AND XIAOHUA YAO
Abstract.
In this paper, we consider the Cauchy global problem for the L -critical semi-linear heat equations ∂ t h = ∆ h ± | h | d h, with h (0 , x ) = h , where h is an unknown realfunction defined on R + × R d . In most of the studies on this subject, the initial data h be-longs to Lebesgue spaces L p ( R d ) for some p ≥ H s ( R d ) with s >
0. We here prove that there exists some positive constant ε depending on d , such thatthe Cauchy problem is locally and globally well-posed for any initial data h which is radial,supported away from origin and in the negative Sobolev space ˙ H − ε ( R d ) including L p ( R d )with certain p < L -estimateboth as time t → t → + ∞ were considered. Introduction
Consider the initial value problem for a semilinear heat equation: ( ∂ t h = ∆ h ± | h | γ − h,h (0 , x ) = h ( x ) , x ∈ R d , (1.1)where h ( t, x ) is an unknown real function defined on R + × R d , d ≥ γ >
1. The positive sign“+” in nonlinear term of (1.1) denotes focusing source, and the negative sign “-” denotes thedefocusing one. The Cauchy problem (1.1) has been extensively studied in Lebesgue space L p ( R d ) by many peoples, see e.g. [2, 3, 4, 6, 7, 10, 12, 13, 14, 15, 16, 18, 19, 21, 25, 26] andso on. The equation enjoys an interesting property of scaling invariance h λ ( t, x ) := λ / ( γ − h ( λ t, λx ) , h λ (0 , x ) := λ / ( γ − h ( λx ) , λ > , that is, if h ( t, x ) is the solution of heat equation (1.1), then h λ ( t, x ) also does with the scalingdata λ /γ h ( λx ). An important fact is that Lebsgue space L p c ( R d ) with p c = d ( γ − is theonly one invariant under the same scaling transform: h ( x ) λ / ( γ − h ( λx ) . If we consider the initial data h ∈ L p ( R d ), then the scaling index p c = d ( γ − Mathematics Subject Classification. plays a critical role on the local/global well-posedness of (1.1). Roughly speaking, one candivide the dynamics of (1.1) into the following three different regimes: (A) the subcritialcase p > p c , (B) the critical case p = p c , (C) the supercritical case p < p c . Specifically,In cases (A) and (B), i.e. p ≥ p c , when p > γ , Weissler in [25] proved the local existenceand uniqueness of solution h ∈ C ([0 , T ); L q ( R d )) ∩ L ∞ loc ((0 , T ]; L ∞ ( R d )). Later, Brezis andCazenave [2] proved the unconditional uniquessness of Weissler’s solution. In double criticalcase p = p c = γ ( i.e. p = γ = dd − ), the local conditional wellposedness of the problem(1.1) was due to Weissler in [26], but the unconditional uniqueness fails, see Ni-Sacks [16],Terraneo [22]. In the supercritical case ( C ), i.e. p < p c , it seems that there exists no localsolution in any reasonable sense for some initial data h ∈ L p ( R d ). In particular, in focusingcase, there exists a nonnegative function h ∈ L p ( R d ) such that the (1.1) does not admitany nonnegative classical L p -solution in [0 , T ) for any T >
0, see e.g. Brezis and Cabr´e[1], Brezis and Cazenave [2], Haraux-Weissler[9] and Weissler [25, 26]. Also, one see bookQuitnner-Souplet[17] for many related topics and references.In this paper, we mainly concerned with the local and global existence of solution forsome supercritical initial data h ∈ L p ( R d ) by p < p c and more generally, initial data in˙ H − ǫ . For simplicity, we only consider the Cauchy problem for the L -critical semilinear heatequations, ∂ t h = ∆ h + µ | h | d h,h (0 , x ) = h ( x ) , x ∈ R d , (1.2)That is, p c = 2 ( i.e γ = 1 + d ), we will prove that there exists some positive constant ε depending on d , such that the Cauchy problem is locally and globally wellposed for any initialdata h is radial, supported away from origin and in the negative Sobolev space ˙ H − ε ( R d ),which includes certain L p -space with p < p c = 2 as a subspace (see Remark 1.1 below). Weremark that, at present the the range of ǫ in the following theorem may not be optimalto local and global existence of solution of the problem (1.2). On the other hand, we alsomention that a result in Brezis and Freidman[3] implies that the problem (1.2) has no anysolution ( even weak one) with a Dirac initial data δ , which is in H − ǫ ( R d ) for any s > d/ Theorem 1.1.
Let µ = ± and ε ∈ h , d − d + 2 (cid:17) , d ≥ . (1.3) LOBAL ROUGH SOLUTION FOR SEMILINEAR HEAT EQUATION 3
Suppose that h ∈ ˙ H − ε ( R d ) is a radial initial data satisfying supp h ⊂ { x : | x | ≥ } . Thenthere exists a time δ = δ ( h ) > and a unique strong solution h ∈ C ([0 , δ ); L ( R d ) + ˙ H − ε ( R d )) ∩ L d ) d tx ([0 , δ ] × R d ) to the equation (1.2) with the initial data h . Moreover, the following two statements hold: (1). If d > , then the solution h is unique in the following sense that there exists a uniquefunction w in C ([0 , δ ] , L ( R d )) such that h = e t ∆ h + w. (1.4)(2). If k h k ˙ H − ε ( R d ) is small enough, then the solution is global in time and satisfies thefollowing decay estimate for d ≥ , k h ( t ) k L . t − ε k h k ˙ H − ε , t > . Remark 1.1. If h ∈ L p for some p < , then there exists some ǫ > such that h ∈ ˙ H − ε ( R d ) and k h k ˙ H − ε ( R d ) . k h k L p ( R d ) by the Sobolev embedding estimate ( see e.g. Lemma 3.1 below ). Thus, Theorem 1.1 showsthat the solution h of the equation (1.2) exists locally for any radial and supported away fromzero initial datum h in L p ( R d ) as p ∈ (cid:16) d +4 d − d +2 d , (cid:17) and d ≥ . Remark 1.2.
It seems that the restriction d > is necessary for unconditional uniqueness.In fact, when d = 4 , the uniqueness problem is related to the “double critical” case ( i.e. p = p c = γ = dd − = 2 ). It was well-known that the unconditional uniqueness failed byNi-Sacks [16] and Brezis and Cazenave [2] . Finally, it is worth mentioning that in the defocusing case, the smallness restriction onthe initial datum in the statement (2) is not necessary for global existence. Indeed, we have h ( δ ) ∈ L ( R d ), then it follows by considering the solution from t = δ . Moreover, it is easy tofind a large class of h satisfying the conditions of theorem above. As described in Remark1.1, our result shows that the solution h of the equation (1.2) exists globally on R +, for anythe initial datum h in L p ( R d ) with some p <
2, which is radial and supported away fromzero.The paper is organized as follows: In Section 2, we will list several useful lemmas aboutLittlewood-Paley theory, and space-time estimates for the solution of linear heat equation.Then in Section 3, we will give the proof of the main results, respectively.
AVY SOFFER, YIFEI WU, AND XIAOHUA YAO Preliminary
Littlewood-Paley multipliers and related inequalities.
Throughout this paper,we write A . B to signify that there exists a constant c such that A ≤ cB , while we denote A ∼ B when A . B . A . We first define the Littlewood-Paley projection multiplier . Let ϕ ( ξ ) be a fixed real-valued radially symmetric bump function adapted to the ball { ξ ∈ R d : | ξ | ≤ } which equals 1 on the ball { ξ ∈ R d : | ξ | ≤ } . Define a dyadic number to anynumber N ∈ Z of the form N = 2 j where j ∈ Z ( the integer set). For each dyadic number N , we define the the Fourier multipliers \ P ≤ N f ( ξ ) := ϕ ( ξ/N ) ˆ f ( ξ ) , d P N f ( ξ ) := ϕ ( ξ/N ) − ϕ (2 ξ/N ) ˆ f ( ξ ) , where ˆ f denotes the Fourier transform of f . Moreover, define P >N = I − P ≤ N and P
Let e t ∆ denote the heat semigroupon R d . Then for suitable function f , e t ∆ f solves the linear heat equation ∂ t h = ∆ h, h (0 , x ) = f ( x ) , t > , x ∈ R d , and the solution satisfies the following fundamental space-time estimates: LOBAL ROUGH SOLUTION FOR SEMILINEAR HEAT EQUATION 5
Lemma 2.2.
Let f ∈ L p ( R d ) for ≤ p ≤ ∞ , then (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) L ∞ t L px ( R + × R d ) . k f k L p ( R d ) . (2.1) Moreover, let I ⊂ R + , then for f ∈ L ( R d ) and F ∈ L d ) d +4 tx ( R + × R d ) , (cid:13)(cid:13) ∇ e t ∆ f (cid:13)(cid:13) L tx ( R + × R d ) . k f k L ( R d ) ; (2.2) (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) L d ) dtx ( R + × R d ) . k f k L ( R d ) ; (2.3) (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ F ( s ) ds (cid:13)(cid:13)(cid:13) L ∞ t L x ∩ L d ) dtx ∩ L t ˙ H x ( I × R d ) . k F k L d ) d +4 tx ( R + × R d ) . (2.4)We can give some remarks on the inequalities (2.1) − (2.4) above as follows:(i). The estimate (2.1) is classical and immediately follows from the Younger inequalityby the following heat kernel integral:( e t ∆ f )( x ) = (4 πt ) − d/ Z R d e −| x − y | / t f ( y ) dy, t > . More generally, for all 1 ≤ p ≤ q ≤ ∞ , the following (decay) estimates hold: k e t ∆ f k L q ( R d ) . t d ( q − p ) k f k L p ( R d ) , t > . (2.5)(ii). The estimate (2.2) is equivalent to a kind of square-function inequality on L ( R d ),which can be reformulated as (cid:13)(cid:13)(cid:13)(cid:16) Z ∞ |√ t ∇ e t ∆ f | dtt (cid:17) (cid:13)(cid:13)(cid:13) L ( R d ) . k f k L ( R d ) , which follows directly by the Plancherel’s theorem, and also holds in the L p ( R d ) for 1 < p < ∞ ( see e.g. Stein[20, p. 27-46] ).(iii). The estimate (2.3) can be obtained by interpolation between the (2.1) and (2.2): (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) L d ) dtx ( R + × R d ) . (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) d +2 L ∞ t L x ( R + × R d ) (cid:13)(cid:13) ∇ e t ∆ f (cid:13)(cid:13) dd +2 L tx ( R + × R d ) . (iv). The estimate (2.4) consists of the three same type inequalities with the differentnorms L ∞ t L x , L d ) d tx and L t ˙ H x on the left side. As shown in (iii) above, the second norm L d ) d tx can be controlled by interpolation between L ∞ t L x and L t ˙ H x . Because of similarity oftheir proofs, we can give a proof to the first one, which is the special case of the followinglemma. It is worth to noting that when p < ∞ , the estimate is L -subcritical. Lemma 2.3.
Let ≤ p ≤ ∞ , and the pair ( p , r ) satisfy p + dr = d p , ≤ p ≤ , < r ≤ , AVY SOFFER, YIFEI WU, AND XIAOHUA YAO then (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ F ( s ) ds (cid:13)(cid:13)(cid:13) L pt L x ( R + × R d ) . k F k L p t L r x ( R + × R d ) . Proof.
By Plancherel’s theorem, it is equivalent that (cid:13)(cid:13)(cid:13) Z t e − ( t − s ) | ξ | b F ( ξ, s ) ds (cid:13)(cid:13)(cid:13) L pt L ξ ( R + × R d ) . k F k L p t L r x ( R + × R d ) . (2.6)Since by the Young inequality of the convolution on R + , for any 1 ≤ p ≤ p ≤ ∞ , (cid:13)(cid:13)(cid:13) Z t e − ( t − s ) | ξ | b F ( ξ, s ) ds (cid:13)(cid:13)(cid:13) L p ( R + ) . (cid:13)(cid:13)(cid:13) | ξ | − ( p + p ′ ) b F ( ξ, · ) (cid:13)(cid:13)(cid:13) L p t ( R + ) . Note that p ≤ ≤ p , thus by Minkowski’s inequality, Plancherel’s theorem, Sobolev’sembedding we obtain (cid:13)(cid:13)(cid:13) Z t e − ( t − s ) | ξ | b F ( ξ, s ) ds (cid:13)(cid:13)(cid:13) L pt L ξ ( R + × R d ) . (cid:13)(cid:13)(cid:13) | ξ | − ( p + p ′ ) b F ( ξ, · ) (cid:13)(cid:13)(cid:13) L ξ L p t ( R + × R d ) . (cid:13)(cid:13)(cid:13) |∇| − ( p + p ′ ) F (cid:13)(cid:13)(cid:13) L p t L x ( R + × R d ) . k F k L p t L r x ( R + × R d ) . which gives the desired estimate (2.6). (cid:3) Finally, we also need the following maximal L p -regularity result for the heat flow. SeeLemarie-Rieusset’s book [5, P.64] for example. Lemma 2.4.
Let p ∈ (1 , ∞ ) , q ∈ (1 , ∞ ) , and let T ∈ (0 , ∞ ] , then the operator A defined by f ( t, x ) Z t e ( t − s )∆ ∆ f ( s, · ) ds is bounded from L p ((0 , T ) , L q ( R d )) to L p ((0 , T ) , L q ( R d )) . Proof of Theorem 1.1
In this section, we will divide several subsection to finish the proof of Theorem 1.1. Forthe end, we first establish a supercritical estimate on the linear heat flow in the followingsubsection.3.1.
A supercritical estimate on the linear heat flow.
Let us recall the following radialSobolev embedding, see [24] for example.
Lemma 3.1.
Let α, q, p, s be the parameters which satisfy α > − dq ; 1 q ≤ p ≤ q + s ; 1 ≤ p, q ≤ ∞ ; 0 < s < d with α + s = d ( 1 p − q ) . LOBAL ROUGH SOLUTION FOR SEMILINEAR HEAT EQUATION 7
Moreover, let at most one of the following equalities hold: p = 1 , p = ∞ , q = 1 , q = ∞ , p = 1 q + s. Then the radial Sobolev embedding inequality holds: (cid:13)(cid:13) | x | α u (cid:13)(cid:13) L q ( R d ) . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) L p ( R d ) , Lemma 3.2.
For any q > and any γ ∈ (cid:0) − q , − q (cid:1) , suppose that the radial function f ∈ H γ ( R d ) satisfying supp f ⊂ { x : | x | ≥ } , then (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) L qtx ( R + × R d ) . (cid:13)(cid:13) |∇| γ f (cid:13)(cid:13) L x ( R d ) . Proof.
By Lemma 2.1, we have (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) L ∞ tx ( R + × R d ) . k f k L ∞ ( R d ) . Let α = d − s > s ∈ ( , k f k L ∞ ( R d ) . k| x | α f k L ∞ ( R d ) . (cid:13)(cid:13) |∇| s f (cid:13)(cid:13) L ( R d ) , where the first inequality above has used the condition supp f ⊂ { x : | x | ≥ } . Thus we getthat (cid:13)(cid:13) e t ∆ f (cid:13)(cid:13) L ∞ tx ( R + × R d ) . (cid:13)(cid:13) |∇| s f (cid:13)(cid:13) L ( R d ) . (3.1)Interpolation between this last estimate and (2.2), gives our desired estimates. (cid:3) Local theory and global criterion.
We use χ ≤ a for a ∈ R + to denote the smoothfunction χ ≤ a ( x ) = , | x | ≤ a, , | x | ≥ a, and set χ ≥ a = 1 − χ ≤ a .Now write h = v + w , (3.2)where v = χ ≥ (cid:0) P ≥ N h (cid:1) , w = h − v . Then we will first claim that w ∈ L ( R d ), and k w k L ( R d ) . N ε (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . (3.3) AVY SOFFER, YIFEI WU, AND XIAOHUA YAO
Note that w = χ ≤ (cid:0) P ≥ N h (cid:1) + P Let h be the function satisfying the hypothesis in Theorem 1.1, then (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . N − (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . (3.4) Proof. By the support property of h , we may write χ ≤ (cid:0) P ≥ N h (cid:1) = χ ≤ (cid:0) P ≥ N χ ≥ h (cid:1) = χ ≤ (cid:0) P ≥ N χ ≥ P ≤ N h (cid:1) + ∞ X M =4 N χ ≤ P ≥ N (cid:0) χ ≥ P M h (cid:1) . (3.5)By Lemma 2.1 and Bernstein’s inequality, we have (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N χ ≥ P ≤ N h (cid:1)(cid:13)(cid:13) L ( R d ) . N − (cid:13)(cid:13) P ≤ N h (cid:13)(cid:13) L ( R d ) . N − (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . (3.6)Moreover, since P ≥ N = I − P LOBAL ROUGH SOLUTION FOR SEMILINEAR HEAT EQUATION 9 Second, we claim that (cid:13)(cid:13) v (cid:13)(cid:13) ˙ H − ε ( R d ) . (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . (3.8)Indeed, (cid:13)(cid:13) v (cid:13)(cid:13) ˙ H − ε ( R d ) . (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) + (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) ˙ H − ε ( R d ) . Hence, we only consider the latter term. By Sobolev’s embedding and H¨older’s inequality,we have (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) ˙ H − ε ( R d ) . (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . Hence (3.8) follows from Lemma 3.3.We denote v L ( t ) = e t ∆ v . Then v L is globally existence, and by Plancherel’s theorem and (3.8) (cid:13)(cid:13) v L ( t ) (cid:13)(cid:13) L ∞ t ˙ H − ε x ( R + × R d ) . (cid:13)(cid:13) v (cid:13)(cid:13) ˙ H − ε ( R d ) . (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) , (3.9)Moreover, let ǫ be a sufficiently small positive constant, then we claim that (cid:13)(cid:13) v L ( t ) (cid:13)(cid:13) L d ) dtx ( R + × R d ) . N − d − d +2 + ε + ǫ (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . (3.10)Indeed, let γ = − d − d +2 + ǫ , then by Lemma 3.2, (cid:13)(cid:13) v L ( t ) (cid:13)(cid:13) L d ) dtx ( R + × R d ) . (cid:13)(cid:13) |∇| γ χ ≥ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . Note that (cid:13)(cid:13) |∇| γ χ ≥ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) ≤ (cid:13)(cid:13) |∇| γ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) + (cid:13)(cid:13) |∇| γ χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . For the former term, since γ < − ε , by Bernstein’s inequality, (cid:13)(cid:13) |∇| γ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . N γ + ε (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . So we only need to estimate the latter term. Let q be the parameter satisfying1 q = 12 − γd , then q > 1. Since γ < 0, by Sobolev’s and H¨older’s inequalities, (cid:13)(cid:13) |∇| γ χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L q ( R d ) . (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . Furthermore, by Lemma 3.3, (cid:13)(cid:13) χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . N − (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . Combining the last two estimates above, we obtain (cid:13)(cid:13) |∇| γ χ ≤ (cid:0) P ≥ N h (cid:1)(cid:13)(cid:13) L ( R d ) . N − (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) . This gives (3.10).Now we denote w = h − v L , then w is the solution of the following equation, ∂ t w = ∆ w ± | h | d h,w (0 , x ) = w ( x ) = h − v . (3.11)The following lemma is the local well-posedness and global criterion of the Cauchy problem(3.11). Lemma 3.4. There exists δ > , such that for any h satisfying the hypothesis in Theorem1.1 and w = h − v , the Cauchy problem (3.11) is well-posed on the time interval [0 , δ ] ,and the solution w ∈ C t L x ([0 , δ ] × R d ) ∩ L d ) d tx ([0 , δ ] × R d ) ∩ L t ˙ H x ([0 , δ ] × R d ) . Furthermore, let T ∗ be the maximal lifespan, and suppose that w ∈ L d ) d tx ([0 , T ∗ ) × R d ) , then T ∗ = + ∞ . In particular, if k h k ˙ H − ε ( R d ) ≪ , then T ∗ = + ∞ .Proof. For local well-posedness, we only show that the solution w ∈ L ∞ t L x ([0 , δ ] × R d ) ∩ L d ) d tx ([0 , δ ] × R d ) ∩ L t ˙ H x ([0 , δ ] × R d ) for some δ > 0. Indeed, the local well-posedness withthe lifespan [0 , δ ) is then followed by the standard fixed point argument. By Duhamel’sformula, we have w ( t ) = e t ∆ w ± Z t e ( t − s )∆ | h ( s ) | d h ( s ) ds. Then by Lemma 2.2, for any t ∗ ≤ δ , (cid:13)(cid:13) w (cid:13)(cid:13) L d ) dtx ([0 ,t ∗ ] × R d ) . k e t ∆ w k L d ) dtx ([0 ,t ∗ ] × R d ) + (cid:13)(cid:13) | h | d h (cid:13)(cid:13) L d ) d +4 tx ([0 ,t ∗ ] × R d ) . k e t ∆ w k L d ) dtx ([0 ,δ ] × R d ) + (cid:13)(cid:13) h (cid:13)(cid:13) d +1 L d ) dtx ([0 ,t ∗ ] × R d ) . Note that (cid:13)(cid:13) h (cid:13)(cid:13) L d ) dtx ([0 ,t ∗ ] × R d ) . (cid:13)(cid:13) v L (cid:13)(cid:13) L d ) dtx ( R + × R d ) + (cid:13)(cid:13) w (cid:13)(cid:13) L d ) dtx ([0 ,t ∗ ] × R d ) , let η = ( d + 1) (cid:0) d − d +2 − ε − ǫ (cid:1) > 0, then using (3.10), we obtain (cid:13)(cid:13) w (cid:13)(cid:13) L d ) dtx ([0 ,t ∗ ] × R d ) . k e t ∆ w k L d ) dtx ([0 ,δ ] × R d ) + N − η (cid:13)(cid:13) h (cid:13)(cid:13) d +1˙ H − ε ( R d ) + (cid:13)(cid:13) w (cid:13)(cid:13) d +1 L d ) dtx ([0 ,t ∗ ] × R d ) . LOBAL ROUGH SOLUTION FOR SEMILINEAR HEAT EQUATION 11 Noting that either k h k ˙ H − ε ( R d ) ≪ 1, or choosing δ small enough and N large enough, wehave k e t ∆ w k L d ) dtx ([0 ,δ ] × R d ) + N − η (cid:13)(cid:13) h (cid:13)(cid:13) d +1˙ H − ε ( R d ) ≪ , then by the continuity argument, we (cid:13)(cid:13) w (cid:13)(cid:13) L d ) dtx ([0 ,δ ] × R d ) . k e t ∆ w k L d ) dtx ([0 ,δ ] × R d ) + N − η (cid:13)(cid:13) h (cid:13)(cid:13) d +1˙ H − ε ( R d ) . (3.12)Further, by Lemma 2.2 again, (cid:13)(cid:13) w (cid:13)(cid:13) L t ˙ H x ([0 ,δ ] × R d ) + sup t ∈ [0 ,δ ] (cid:13)(cid:13) w (cid:13)(cid:13) L x ( R d ) . k w k L x ( R d ) + (cid:13)(cid:13) | h | d h (cid:13)(cid:13) L d ) d +4 tx ([0 ,δ ] × R d ) . k w k L x ( R d ) + (cid:13)(cid:13) v L (cid:13)(cid:13) d +1 L d ) dtx ([0 ,δ ] × R d ) + (cid:13)(cid:13) w (cid:13)(cid:13) d +1 L d ) dtx ([0 ,δ ] × R d ) . Hence, using (3.10) and (3.12), we obtain (cid:13)(cid:13) w (cid:13)(cid:13) L t ˙ H x ([0 ,δ ] × R d ) + sup t ∈ [0 ,δ ] (cid:13)(cid:13) w (cid:13)(cid:13) L x ( R d ) ≤ C, for some C = C ( N, (cid:13)(cid:13) h (cid:13)(cid:13) ˙ H − ε ( R d ) ) > w ∈ L d ) d tx ([0 , T ∗ ) × R d ) , then if T ∗ < + ∞ , we have (cid:13)(cid:13) w ( T ∗ ) (cid:13)(cid:13) L x ( R d ) . k e t ∆ w k L d ) dtx ([0 ,T ∗ ] × R d ) + (cid:13)(cid:13) | h | d h (cid:13)(cid:13) L d ) d +4 tx ([0 ,T ∗ ) × R d ) . k w k L x ( R d ) + N − η (cid:13)(cid:13) h (cid:13)(cid:13) d +1˙ H − ε ( R d ) + (cid:13)(cid:13) w (cid:13)(cid:13) d +1 L d ) dtx ([0 ,T ∗ ) × R d ) . Hence, w exists on [0 , T ∗ ], and w ( T ∗ ) ∈ L ( R d ). Hence, using the local theory obtainedbefore from time T ∗ , the lifespan can be extended to T ∗ + δ , this is contradicted with thedefinition of the maximal lifespan T ∗ . Hence, T ∗ = + ∞ . (cid:3) Uniqueness. Here we adopt the argument in [15], where the main tool is the themaximal L p -regularity of the heat flow. Let h , h be two distinct solutions of (1.2) with thesame initial data h , and write h = e s ∆ h + w ; h = e s ∆ h + w . By the Duhamel formula, we have w ( t ) = Z t e ( t − s )∆ | e s ∆ h + w | d (cid:0) e s ∆ h + w (cid:1) ds ; w ( t ) = Z t e ( t − s )∆ | e s ∆ h + w | d (cid:0) e s ∆ h + w (cid:1) ds. Denote w = w − w , then w obeys w ( t ) = Z t e ( t − s )∆ h | e s ∆ h + w | d (cid:0) e s ∆ h + w (cid:1) − | e s ∆ h + w | d (cid:0) e s ∆ h + w (cid:1)i ds. Note that there exists an absolute constant C > (cid:12)(cid:12)(cid:12) | e s ∆ h + w | d (cid:0) e s ∆ h + w (cid:1) − | e s ∆ h + w | d (cid:0) e s ∆ h + w (cid:1)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) | e s ∆ h | d + | w | d + | w | d (cid:17) | w | . Then by the positivity of the heat kernel, we have | w ( t ) | ≤ C Z t e ( t − s )∆ (cid:16) | e s ∆ h | d + | w ( s ) | d + | w ( s ) | d (cid:17) | w ( s ) | ds. Then we get that for 2 ≤ p < ∞ , τ ∈ (0 , δ ], k w k L pt ((0 ,τ ); L ( R d )) . (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ | e s ∆ h | d | w ( s ) | ds (cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) + (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ (cid:16) | w ( s ) | d + | w ( s ) | d (cid:17) | w ( s ) | ds (cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . For the first term in the right-hand side above, using Lemma 2.3 and choosing p large enough,we have (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ | e s ∆ h | d | w ( s ) | ds (cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . (cid:13)(cid:13)(cid:13) | e s ∆ h | d | w ( s ) | (cid:13)(cid:13)(cid:13) L p t ((0 ,τ ); L r ( R d )) , where we have chose ( p , r ) that1 p = 2 d + 2 + 1 p ; 1 r = 2 d + 2 + 12 . (Note that d > p is large, we have that p ∈ (1 , , r ∈ (1 , (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ | e s ∆ h | d | w ( s ) | ds (cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . (cid:13)(cid:13)(cid:13) | e s ∆ h | d | w ( s ) | (cid:13)(cid:13)(cid:13) L p t ((0 ,τ ); L r ( R d )) . (cid:13)(cid:13) e s ∆ h (cid:13)(cid:13) d L d +2) dtx (cid:0) (0 ,τ ) × R d (cid:1)(cid:13)(cid:13) w (cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . For the second term in the right-hand side above, using Lemma 2.4, (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ (cid:16) | w ( s ) | d + | w ( s ) | d (cid:17) | w ( s ) | ds (cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . (cid:13)(cid:13)(cid:13) ( − ∆) − (cid:16)(cid:0) | w ( s ) | d + | w ( s ) | d (cid:1) | w ( s ) | (cid:17)(cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . LOBAL ROUGH SOLUTION FOR SEMILINEAR HEAT EQUATION 13 Since d > 4, by Sobolev’s embedding, we further have (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ (cid:16) | w ( s ) | d + | w ( s ) | d (cid:17) | w ( s ) | ds (cid:13)(cid:13)(cid:13) L pt ((0 ,τ ); L ( R d )) . (cid:13)(cid:13)(cid:0) | w ( s ) | d + | w ( s ) | d (cid:1) | w ( s ) | (cid:13)(cid:13) L pt ((0 ,τ ); L dd +4 ( R d )) . (cid:0) k w k d L ∞ t ((0 ,τ ); L ( R d )) + k w k d L ∞ t ((0 ,τ ); L ( R d )) (cid:1) k w k L pt ((0 ,τ ); L ( R d )) . Collection the estimates above, we obtain that k w k L pt ((0 ,τ ); L ( R d )) . ρ ( τ ) · k w k L pt ((0 ,τ ); L ( R d )) , (3.13)where ρ ( τ ) = (cid:13)(cid:13) e s ∆ h (cid:13)(cid:13) d L d +2) dtx (cid:0) (0 ,τ ) × R d (cid:1) + k w k d L ∞ t ((0 ,τ ); L ( R d )) + k w k d L ∞ t ((0 ,τ ); L ( R d )) . By (3.10) and Lemma 2.2, we have (cid:13)(cid:13) e s ∆ h (cid:13)(cid:13) L d +2) dtx (cid:0) (0 ,τ ) × R d (cid:1) → , when τ → . Further, since w , w ∈ C ([0 , δ ] , L ( R d )), we getlim τ → ρ ( τ ) → . Hence, choosing τ small enough and from (3.13), we obtain that w ≡ t ∈ [0 , τ ). Byiteration, we have w ≡ w on [0 , δ ]. This proves the first statement (1) in Theorem 1.1.3.4. L -estimates. In this subsection, we prove the second statement (2) in Theorem 1.1.Firstly, by Lemma 3.4, when k h k ˙ H − ε ( R d ) ≪ 1, we immediately have the global existenceof the solution for the both cases µ = ± 1. However, in the defocusing case ( µ = 1). thesmallness of k h k ˙ H − ε ( R d ) ≪ h = v L + w and (cid:13)(cid:13) v L (cid:13)(cid:13) L ( R d ) = (cid:13)(cid:13) e − t | ξ | b v ( ξ ) (cid:13)(cid:13) L ξ ( R d ) . (cid:13)(cid:13) e − t | ξ | | ξ | ε (cid:13)(cid:13) L ∞ ξ ( R d ) k v k ˙ H − ε . t − ε k h k ˙ H − ε . (3.14)Hence, from Lemma 3.4, we have h ( δ ) ∈ L ( R d ). Let I = [0 , T ∗ ) be the maximal lifespan ofthe solution h of the Cauchy problem (1.2). Then from the L estimate of the solution (byinner producing with h in (1.2)), we havesup t ∈ I k h k L + k∇ h k L tx ( I × R d ) ≤ k h k L . This gives the uniform boundedness of (cid:13)(cid:13) h (cid:13)(cid:13) L d ) dtx ( I × R d ) and thus (cid:13)(cid:13) w (cid:13)(cid:13) L d ) dtx ( I × R d ) . Then bythe global criteria given in Lemma 3.4, we have T ∗ = + ∞ . Secondly, we consider the time estimate of the solution ( µ = ± t ≤ 1, it followsfrom (3.14) and Lemma 3.4, that k h ( t ) k L . t − ε k h k ˙ H − ε , for any t ∈ (0 , . So it remains to show the decay estimate when t > 1. By Duhamel’s formula, we have k h ( t ) k L ( R d ) ≤ (cid:13)(cid:13) e t ∆ h (cid:13)(cid:13) L ( R d ) + (cid:13)(cid:13)(cid:13) Z t e ( t − s )∆ | h ( s ) | d h ( s ) ds (cid:13)(cid:13)(cid:13) L ( R d ) . Similar as (3.14), we have (cid:13)(cid:13) e t ∆ h (cid:13)(cid:13) L ( R d ) . t − ε k h k ˙ H − ε . Then using the estimate above and Lemma 2.5, we further have k h ( t ) k L ( R d ) . t − ε k h k ˙ H − ε + Z t (cid:13)(cid:13)(cid:13) e ( t − s )∆ | h ( s ) | d h ( s ) (cid:13)(cid:13)(cid:13) L ( R d ) ds . t − ε k h k ˙ H − ε + Z t | t − s | − (cid:13)(cid:13)(cid:13) | h ( s ) | d h ( s ) (cid:13)(cid:13)(cid:13) L dd +4 ( R d ) ds . t − ε k h k ˙ H − ε + Z t | t − s | − k h k d +1 L ( R d ) ds. In the last step we have used the fact d ≥ dd +4 ≥ k h k X ( T ) = sup t ∈ [0 ,T ] (cid:16) t ε k h ( t ) k L ( R d ) (cid:17) . Fixing T > 1, then for any t ∈ (1 , T ], k h ( t ) k L ( R d ) . t − ε k h k ˙ H − ε + Z t | t − s | − s − ε ( d +1) ds k h ( t ) k d +1 X ( T ) . t − ε k h k ˙ H − ε + t − ε ( d +1)+ k h ( t ) k d +1 X ( T ) . t − ε (cid:16) k h k ˙ H − ε + k h ( t ) k d +1 X ( T ) (cid:17) . Thus we obtain that k h ( t ) k X ( T ) . k h k ˙ H − ε + k h ( t ) k d +1 X ( T ) . By the continuity argument, we get k h ( t ) k X ( T ) . k h k ˙ H − ε . 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