# Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

Authors: Lihua Deng; Xianguang Shang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.


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## Body

## 1. Introduction and Statement of the Main Results

We consider the nonnegative solutions of the following Cauchy problem:(1)ut−divDump−2Dum=at−σuqDul,inST=RN×0,T,(2)ux,0=μ,onRN,where m>0, p>1, mp−1>1, a>0, −∞<σ<p−l/p, q≥0, 0<l<p, T>0, and N≥1 and µ is a nonnegative Radon measure. Equation (1) has been intensively investigated in the last decades because of both its mathematical interest and its potential for applications. In particular, it has been proposed as an appropriate model for surface growth by ballistic deposition and specifically for vapour deposition and the sputter deposition of thin films of aluminium and rare earth metals. One can refer to the bibliographies [1–7] and the references therein for more details on the physical situations.For the homogeneous casea=0, the existence of solutions and the initial trace problem were studied in [8–10] for the case p=2. As m=1, DiBenedetto and Herrero [11] established the existence of solutions under optimal assumptions on the initial data and initial trace for nonnegative solutions. When m≠1 and p≠2, one can refer to [12–14] for the existence of solutions, initial trace of nonnegative solutions, and the weak Harnack inequalities for weak supersolutions, respectively.Concerning the caseσ=0, the existence of solutions and initial trace of nonnegative solutions to the problem (1) and (2) have been studied extensively (see, e.g., [15–27]). By a priori estimates and the compactness methods, Andreucci [18] proved the existence of solutions of (1) with p=2 under optimal assumptions on the initial data. Later, Chen and Zhao [22] and Shang and Li [27] separately extended the results of [18] to the Cauchy problem (1) and (2) with m=1, where the initial data are measured. Recently, Shang and Cheng [25] established the local existence of solutions to the Cauchy problem (1) and (2) with mp−1>1 and initial data in LlocrRN with r≥1. However, the existence of solutions for initial data measures was left open. In the bounded domain, the critical extinction exponent, the existence, and uniqueness of weak solutions were studied [28, 29]. Weng [30] established the existence and stability of weak solutions to the double degenerate evolutionary px-Laplacian equation.Asσ>0, as well as we know there are few results in this direction. For the case m=1, p=2, and l=0, the existence and nonexistence of solutions for (1) and (2) were obtained by Meier [31]. Later, based on the a priori estimates and the improved De Giorgi iteration methods, Andreucci [17] established the local and global existence and nonexistence of solutions to (1) and (2) with p=2 and l=0 and initial data in LlocrRN with r≥1. Recently, for the case m=1 and l=0, the local existence of solutions to the Cauchy problem (1) and (2) with initial data measures was studied by Shang [24].Here we consider the existence of solutions to the Cauchy problem (1) and (2) in the spirit of [17, 18, 24, 25]. Due to the more complicated structure of the equation and lower regularity of the initial data, the existence issue considered here becomes more difficult. This makes it harder to get the uniform a priori L∞-estimates and gradient estimates. Fortunately, using delicate estimates, we can overcome these difficulties and establish the local existence of solutions under optimal assumptions on the initial data.As preparations, we first state several notations which will be used frequently later.Definition 1.
A nonnegative measurable functionux,t defined in ST is called a weak solution of (1) and (2), if(3)u∈Cloc0,T;Lloc1RN∩Lloc∞ST,um∈Llocp0,T;Wloc1,pRN,t−σuqDul∈Lloc1ST,∫0t∫RN−uφτ+Dump−2Dum⋅Dφdxdτ=∫0t∫RNaτ−σuqDulφdxdτ,for all φ∈C01ST. Moreover,(4)limt⟶0∫RNux,tηxdx=∫RNηdμ,∀η∈C01ST.
Weak subsolutions (resp. supersolutions) are defined in the same way except that the= in (3) is replaced by ≤ (resp. ≥) and φ is taken to be nonnegative.
Set(5)μ=supx∈RNsup0<ρ<1ρθ∫Bρxdμ,ut=sup0<τ<tsupx∈RNsup0<ρ<1ρθ∫Bρxuy,τdy,0<t<T,where 0≤θ≤N and(6)∫Edμ=1E∫Edμ,Eis the Lebesgue measure ofE,Bρx=yy−x<ρ.
Moreover, we useγa1,a2,…,an to denote positive constants depending only on specified quantities a1,a2,…,an, which may vary from line to line.
We now state our main existence results as follows.Theorem 1 (the caseq+l≥mp−1).
Letμ be finite and(7)σ+θpq+l−lmp−1−p+lpp+θmp−1−1<p−lp.
Then there exists a solution to (1) and (2) defined in RN×0,T0, where T0=T0μ,N,m,p,a,q,l,σ,θ, such that for all 0<t<T0, we have(8)ut≤γμ,(9)u⋅,t∞,RN≤γt−θ/p+θmp−1−1μp/κ,(10)∫0t∫Bρ/2x0Dump−1dxdτ≤γρN−θt1/p+θmp−1−1μ1+mp−1−1/p+θmp−1−1,(11)∫0t∫Bρ/2x0τ−σuqDuldxdτ≤γρN−θtp−l/p−σ−θpq+l−lmp−1−p+l/pp+θmp−1−1μ1+pq+l−lmp−1−p+l/p+θmp−1−1,where γ=γN,m,p,a,q,l,σ,θ, κ=Nmp−1−1+p, and x0∈RN is fixed.Remark 1.
Condition (7) is actually optimal in the class considered here in some sense (see Theorem 3 below). In fact, if σ+θpq+l−lmp−1−p+l/pp+θmp−1−1>p−l/p, then u0x=x−δχB10x with x∈RN\0 and 0<δ<N can produce a counterexample such that no nonnegative solution to (1) and (2) may exist. This claim can be shown by following the method to prove Proposition 2.1 of [17] and Theorem 3 here, so we omit the details. Moreover, (7) extends the classical conditions in [17, 18] for p=2 and [22, 24, 27] for m=1 to the problem considered here.Theorem 2 (the caseq+l<mp−1).
Letμ be finite. Let pq+l≥lmp−1. Then there exists a solution to (1) and (2) defined in RN×0,T0′, where T0′=T0′μ,N,m,p,a,q,l,σ, such that for all 0<t<T0′, we have(12)ut≤γμ+1,(13)u⋅,t∞,RN≤γt−N/κμp/κ+1,(14)∫0t∫Bρ/2x0τ−σuqDuldxdτ≤γρN−θμ+1+ρNp−lθ/pμ+1l/ptp−l/p−σ−lθr0/pp+θmp−1−1μ+1lr0/κ+tp−l/p−σ−lr0/p+γt1−p−l/pρpN−θ+lθ/pμ+1p−l/p+ρN,(15)∫0t∫Bρ/2x0Dump−1dxdτ≤γρN−θμ+1t1/p+θmp−1−1μ+1mp−1−1/κ+tp−1/pm−p+θmp−1r1/p+θmp−1−1μ+1p−1r1/κ+t1/p−p−1r1/p−θp−1m−r1−1/pp+θmp−1−1μ+1p−1m−r1−1/κ+tmp−1/p+γρNp−θ/pμ+11/pt1−σp−1/p−l−θp−1m−r1−1/pp+θmp−1−1μ+1p−1m−r1−1/κ+tp−1m−r1+1/p−σp−1/p−l,where r0 and r1 depending on N,m,p,a,q,l,σ, and θ are positive constants such that the exponents in (14) and (15) are positive, respectively, γ=γN,m,p,a,q,l,σ,θ and x0∈RN is fixed.Remark 2.
The dependence ofT0 and T0′ on the quantities specified in the statements of Theorems 1 and 2 can be made explicitly. One can refer to the proof of Theorems 1 and 2, respectively.Remark 3.
With minor revisions, one can prove that Theorems1 and 2 also hold for a<0. This paper is mainly devoted to the case of a>0, so we do not give more details for the case a<0. However, we would like to mention many important developments without completeness for theproblem (1) and (2) with a<0; see [32–42] and the references therein, where the existence and nonexistence, quantitative properties, and asymptotic behavior of solutions were studied.
Lastly, we state a result in the direction of the optimality of the critical threshold forθ in (7).Theorem 3.
Letl∈1,+∞. Let u be a nonnegative solution to (1) in RN×0,T such that ut<∞, 0<t<T, and(16)μBρx0≥ϵρN−θ,0<ρ<ϵ,where ϵ>0 and x0∈RN are given. Then σ+θpq+l−lmp−1−p+l/pp+θmp−1−1≤p−l/p.The rest of the paper will be divided into two sections: In Section2, as preparations, we first establish some a priori estimates and then prove Theorems 1 and 2. In sequence, we finish the proof of Theorem 3 in Section 3.


## 2. Proofs of Theorems1 and 2

In this section, we shall establish theL∞-estimates and gradient estimates of solutions which are stated in Lemma 1 and Lemma 2 below, respectively, and thereby give proofs of Theorem 1.1 and Theorem 1.2 based on these key estimates.For technical convenience, here for anyu∈L∞RN×ϵ,T∗ with ϵ>0, we define(17)ut=sup0<τ<tsupx∈RNsupRτ<ρ<1ρθ∫Bρxuy,τdy,Rt=Γt1/p+θmp−1−1,for all 0<t<T∗, and(18)Γ=Cμmp−1−1/p+θmp−1−1,ifq+l≥mp−1,1,ifq+l<mp−1,where C>0 is to be chosen a priori dependent on N,m,p,a,q,l,σ,θ. We also assume that T∗ is chosen so that RT∗≤1. Note that the last assumption is obvious for q+l<mp−1 and is meaningful for q+l≥mp−1 because Γ is independent of T∗.We now turn to the proof of Theorem1 and Theorem 1.2. To achieve these goals, we first state and prove Lemma 1.Lemma 1.
Letu be a nonnegative continuous weak subsolution of (1). Then the following two statements hold:(i)
The caseq+l≥mp−1. Assume that a time 0<T′<T∗ is given such that(19)Γ−ptθmp−1−1/p+θmp−1−1u⋅,t∞,RNmp−1−1+t1−σp/p−lu⋅,t∞,RNpq+l−lmp−1/p−l−1≤1,∀0<t<T′.Then(20)u⋅,t∞,RN≤γt−θ/p+θmp−1−1ΓpN−θ/κutp/κ,∀0<t<T′,whereγ=γN,m,p,a,q,l,σ,θ and κ=Nmp−1−1+p.(ii)
The caseq+l<mp−1. Assume also that a time 0<T″<T∗ is given such that(21)tθmp−1−1/p+θmp−1−1u⋅,t∞,RNmp−1−1+t1−σp/p−l≤1,∀0<t<T″.Then(22)u⋅,t∞,RN≤γt−θ/p+θmp−1−1utp/κ+t,∀0<t<T″,whereγ=γN,m,p,a,q,l,σ,θ.Proof.
We divided the proof into two steps:Step 1. Following the method to prove Lemma 1 in [25], if q+l≥mp−1, we obtain(23)u⋅,t∞,Bρ/2≤γt−N+p/κ∫0t∫Bρudxdτp/κ,for all0<t<T′. If q+l<mp−1, then we have(24)u⋅,t∞,Bρ/2≤γt−N+p/κ∫0t∫Bρudxdτp/κ+t,for all0<t<T″.Step 2. Set ρ=Rt. Then (23) and (17) imply that(25)u⋅,t∞,Bρ/2≤γt−N+p/κ∫0t∫BRtudxdτp/κ≤γt−N/κRpN−θ/κtutp/κ=γt−θ/p+θmp−1−1ΓpN−θ/κutp/κ,for all0<t<T′. Therefore, (20) is obtained. Similarly, (24) and (17) yield (22) for all 0<t<T″.Remark 4.
It follows from the definitions ofut and ut that ut≤ut. Combining Lemma 1 and (17) and (18), we obtain(26)ut≤ut+γCμNmp−1−1/κutp/κ,∀0<t<T′,for the case q+l≥mp−1, and(27)ut≤ut+utp/κ+t1+θ/p+θmp−1−1,∀0<t<T″,for the case q+l<mp−1, where T′ and T″ are as in Lemma 1.
We now start to state and prove Lemma2.Lemma 2.
Letu be a nonnegative continuous weak subsolution of (1). Then for all Rt≤ρ≤1 and Bρx0⊂RN, the following two statements hold:(i)
The caseq+l≥mp−1. Let (19) and (7) hold. Then for all 0<t<T′, we have(28)∫0t∫Bρ/2x0τ−σuqDuldxdτ≤γGttp−l/p−σ−θpq+l−lmp−1−p+l/pp+θmp−1−1ΓN−θutpq+l−lmp−1−p+l/κ,(29)∫0t∫Bρ/2x0Dump−1dxdτ≤γGtt1/p+θmp−1−1ΓN−θutmp−1−1/κ,whereGt=sup0<τ<t∫Bρx0ux,τdx,κ=Nmp−1−1+p and γ=γN,m,p,a,q,l,σ,θ.(ii)
The caseq+l<mp−1. Let (21) hold. Then for all 0<t<T″, we have(30)∫0t∫Bρ/2x0τ−σuqDuldxdτ≤γGt+ρNp−l/pGl/pttp−l/p−σ−lθr0/pp+θmp−1−1utlr0/κ+tp−l/p−σ−lr0/p+γt1−p−l/pρNl/pGp−l/pt+ρN,where r0=r0N,m,p,a,q,l,σ,θ>0 is a constant such that the exponents in (30) are positive and γ=γN,m,p,a,q,l,σ,θ. Moreover,(31)∫0t∫Bρ/2x0Dump−1dxdτ≤γGtt1/p+θmp−1−1utmp−1−1/κ+tp−1/pm−p+θmp−1r1/p+θmp−1−1utp−1r1/κ+t1/p−p−1r1/p−θp−1m−r1−1/pp+θmp−1−1utp−1m−r1−1/κ+tmp−1/p+γρNp−1/pG1/ptt1−σp−1/p−l−θp−1m−r1−1/pp+θmp−1−1utp−1m−r1−1/κ+tp−1m−r1+1/p−σp−1/p−l,where r1=r1N,m,p,a,q,l,σ,θ>0 is a constant such that the exponents in (31) are positive and γ=γN,m,p,a,q,l,σ,θ.Proof.
In the following, we only prove (28) and (30), since (29) and (31) can be proved similarly and we omit the details.
We first prove (28). For notational convenience, we set Bρ=Bρx0. Take φ=tβurζp as a test function in (3), where ζ is a piecewise smooth cut-off function in Bρ, such that 0≤ζ≤1inBρ,ζ≡1inBρ/2, and Dζ≤2/ρ. Moreover, β>0 and r>0 are to be chosen. Then standard calculations imply(32)rmp−1∫0t∫Bρτβum−1p−1+r−1Dupζpdxdτ≤βr+1∫0t∫Bρτβ−1ur+1ζpdxdτ+pmp−1∫0t∫Bρτβum−1p−1+rDup−1Dζp−1dxdτ+∫0t∫Bρaτβ−σuq+rDulζpdxdτ.
Applying Young’s inequality, together with (19) and (20), yield(33)∫0t∫Bρτβum−1p−1+r−1Dupζpdxdτ≤γ∫0t∫Bρτβ−1ur+1dxdτ+γ∫0t∫Bρτβump−1+rDζpdxdτ+γ∫0t∫Bρτβ−σp/p−lupq+l−lmp−1/p−l+rdxdτ≤γ∫0t∫Bρτβ−11+ρ−pτump−1−1+τ1−σp/p−lupq+l−lmp−1/p−l−1ur+1dxdτ≤γGt∫0tτβ−1u⋅,τ∞,Bρrdτ≤γGttβ−θr/p+θmp−1−1ΓpN−θr/κutpr/κ,provided(34)β>θrp+θmp−1−1.
Next, we estimate by Hölder’s inequality(35)∫0t∫Bρτ−σuqDulζldxdτ≤γ∫0t∫Bρτβum−1p−1+r−1Dupζpdxdτl/p∫0t∫Bρτ−σp+βl/p−lupq+l−lmp−1−rl/p−ldxdτp−l/p.
Again using (20), we have(36)∫0t∫Bρτ−σp+βl/p−lupq+l−lmp−1−rl/p−ldxdτ≤Gt∫0tτ−σp+βl/p−lu⋅,τ∞,Bρpq+l−lmp−1/p−l−1−rl/p−ldτ≤γGtt1−σp+βl/p−l−θpq+l−lmp−1/p−l−1−rl/p−l/p+θmp−1−1ΓpN−θ/κutp/κpq+l−lmp−1/p−l−1−rl/p−l,provided(37)pq+l−lmp−1p−l−1≥rlp−l,(38)1−σp+βlp−l−θpq+l−lmp−1/p−l−1−rl/p−lp+θmp−1−1>0.
Inserting (33) and (36) into (35), we obtain (28). It is left to choose β and r such that (34), (37), and (38) hold. This is a trivial task if θ=0. Assume that θ>0, and(39)0<β<p−ll.
Then (34) and (37) are implied by(40)r<minp+θmp−1−1θβ,p−llpq+l−lmp−1p−l−1≜z0.
Note that (38) is equivalent to(41)r>p−llθpq+l−lmp−1/p−l−1p+θmp−1−1−1+σpl+βp+θmp−1−1θ≜z0′.
By virtue of (7) and (39), it is easy to check that z0′<z0. Finally, by noting that z0>0, we can choose r>0 with r∈z0′,z0 such that (34), (37), and (38) hold.
We now turn to prove (30). Comparing with the proof of (28), the main differences here are the estimates (33) and (36). Note that upq+l−lmp−1/p−l+r≤u1+r+1, since q+l<mp−1 implies pq+l−lmp−1/p−l<1, and we have(42)∫0t∫Bρτβum−1p−1+r−1Dupζpdxdτ≤γ∫0t∫Bρτβ−1ur+1dxdτ+γ∫0t∫Bρτβump−1+rDζpdxdτ+γ∫0t∫Bρτβ−σp/p−lupq+l−lmp−1/p−l+rdxdτ≤γ∫0t∫Bρτβ−11+ρ−pτump−1−1+τ1−σp/p−lur+1dxdτ+γρNtβ−σp/p−l+1≤γGt∫0tτβ−1u⋅,τ∞,Bρrdτ+γρNtβ−σp/p−l+1≤γGttβ−θr/p+θmp−1−1utpr/κ+tβ+r+γρNtβ−σp/p−l+1,where we have used (21) and (22) in the third and last inequality, respectively, and we again used (34). Assume that(43)pq+l−lmp−1−rl>0.
Then we obtain(44)∫0t∫Bρτ−σp+βl/p−lupq+l−lmp−1−rl/p−ldxdτ≤∫0t∫Bρτ−σp+βl/p−lu+1dxdτ≤Gtt1−σp+βl/p−l+γρNt1−σp+βl/p−l,where we have used the fact upq+l−lmp−1−rl/p−l≤u+1, since q+l<mp−1 and (43) imply pq+l−lmp−1−rl/p−l<1. Inserting (42) and (44) into (35), we obtain (30). It is left to verify that there exist β>0 and r>0 such that (34) and (43) hold. This can be immediately proved by fixing β as (39) and choosing r>0 sufficiently small.
Based on Lemmas1 and 2, we are ready to prove Theorem 1.Proof of Theorem 1.
Consider the approximating problems:(45)unt−divDunmp−2Dunm=amint−σunqDunl,n,inBn×0,∞,unx,t=0,in∂Bn×0,∞,unx,0=u0nx,onBn,where Bn=x∈RNx<n and u0n∈C0∞RN is nonnegative and has compact support in Bn, which satisfy(46)limn⟶∞∫RNu0nηxdx=∫RNηxdμ,∀η∈C0∞RN,u0n≤γNμ.
By the results of [4, 5, 43] and [44, 45], we can obtain the existence of solutions un for approximation problems (45). Moreover, these solutions are Hölder continuous. Then following the methods to prove Theorem 1 in [25], we can complete the proof of Theorem 1 whence we can show estimates (8)–(11) for u and u0 replaced by un and u0n, with constant γ independent of n. To prove these estimates, we will work with (45) and drop the subscript n.
Define(47)t0=sup0<T′T∗2.3holds.
Choose0<t<t0 and let Bρ=Bρx0, where Rt≤ρ≤1 and x0∈RN. Take ζ as the test function in (45), where ζ is as in the proof of Lemma 2. Direct calculation shows that(48)∫Bρ/2ux,tdx≤∫Bρdμ+2ρ∫0t∫BρDump−1dxdτ+∫0t∫Bρaτ−σuqDuldxdτ.
Multiplying (48) by ρθBρ−1, together with (18), (20), (28), and (29), we obtain(49)ρθ∫Bρ/2ux,tdx≤γ1Nμ+γutC−p+θmp−1−1/κutμmp−1−1/κ+γtp−l/p−σ−θpq+l−lmp−1−p+l/pp+θmp−1−1ΓN−θutpq+l−lmp−1−p+l/κ.for all 0<t<t0 and Rt≤ρ≤1. Note that x0∈RN is arbitrarily chosen, it is immediately seen that(50)ut≤γ1Nμ+γutC−p+θmp−1−1/κutμmp−1−1/κ+Mt,where the meaning of Mt is obvious. Set(51)t1=sup0<t<T∗ut≤4γ1μ,t2=sup0<t<T∗Mt<δ,where δ>0 (small) is to be chosen. Note that t1 and t2 are well defined because the stipulated assumptions make sure that ut is continuous in 0,T∗, and the exponent of t in (51) is positive. Let t3=mint0,t1,t2. Then for 0<t<t3, we have(52)γC−p+θmp−1−1/κutμmp−1−1/κ≤γC−p+θmp−1−1/κ4γ1mp−1−1/κ≤14,provided C is suitably chosen. Then if we choose δ<1/4γ, it follows from (50)(53)ut≤2γ1μ,∀0<t<t3.
By (26) and (53), we get(54)ut≤γμ,∀0<t<t3.
Therefore, (8)–(11) follow from (53), (20), (26), (28), and (29), provided that we can indeed find a quantitative estimates below t3≥T0. We may assume t3<T∗ , since the estimate is otherwise trivial. First we note that (53) implies t3<t1. Next, it is easy to rule out the case t3=t0. In fact,(55)Γ−ptθmp−1−1/p+θmp−1−1u⋅,t∞,RNmp−1−1+t1−σp/p−lu⋅,t∞,RNpq+l−lmp−1/p−l−1≤C−p+θmp−1−1/κutμmp−1−1/κ+t1−σp/p−l−θpq+l−lmp−1−p+l/p−lp+θmp−1−1ΓN−θutppq+l−lmp−1−p+l/p−lκ≤12,where we choose C large enough and δ sufficiently small. Finally, we are left with the task of estimating below t3=t2. This can be accomplished at once, by replacing ut with 2γ1μ in the definition of t2 in (51), owing to (53).
Now we turn to prove Theorem 2, which is similar to the proof of Theorem1. However, we give the details for the convenience of readers.Proof of Theorem 2.
We again consider (45). If we show estimates (12)–(15) for u and u0 replaced by un and u0n, with constant γ independent of n, then Theorem 2 can be proved with similar reason as in Theorem 1. Set(56)t0′=sup0<T″T∗22holds.
Again takingζ as the test function in (45), where ζ is as in the proof of Lemma 2, we again obtain (48). Multiplying (48) by ρθBρ−1, together with (22), (30), and (31), we obtain(57)ρθ∫Bρ/2ux,tdx≤γNμ+γut+ρθp−l/putl/ptp−l/p−σ−lθr0/pp+θmp−1−1utlr0/κ+tp−l/p−σ−lr0/p+γt1−p−l/pρθl/putp−l/p+ρN+γutt1/p+θmp−1−1utmp−1−1/κ+tp−1/pm−p+θmp−1r1/p+θmp−1−1utp−1r1/κ+t1/p−p−1r1/p−θp−1m−r1−1/pp+θmp−1−1utp−1m−r1−1/κ+tmp−1/p+γρθp−1/put1/pt1−σp−1/p−l−θp−1m−r1−1/pp+θmp−1−1utp−1m−r1−1/κ+tp−1m−r1+1/p−σp−1/p−l,for all 0<t<t0′ and Rt≤ρ≤1. Note that x0∈RN is arbitrarily chosen, it is immediately seen that(58)ut≤γNμ+γN1tut+γN2t,where(59)N1t=tp−l/p−σ−lθr0/pp+θmp−1−1utlr0/κ+tp−l/p−σ−lr0/p+t1/p+θmp−1−1utmp−1−1/κ+tp−1/pm−p+θmp−1r1/p+θmp−1−1utp−1r1/κ+t1/p−p−1r1/p−θp−1m−r1−1/pp+θmp−1−1utp−1m−r1−1/κ+tmp−1/p,N2t=ρθp−l/putl/ptp−l/p−σ−lθr0/pp+θmp−1−1utlr0/κ+tp−l/p−σ−lr0/p+t1−p−l/pρθl/putp−l/p+ρN+ρθp−1/put1/pt1−σp−1/p−l−θp−1m−r1−1/pp+θmp−1−1utp−1m−r1−1/κ+tp−1m−r1+1/p−σp−1/p−l.
Set(60)t1′=sup0<tT∗N1t+N2t<ϵ,where ϵ>0 (small) is to be chosen. Note that the exponents of t in (60) are positive. Then using (22), (58), and (60) and following the method to fix T0 in Theorem 1, we can obtain a positive time T0′ such that(61)ut≤γμ+1,∀0<t<T0′.
Therefore, (12)–(15) follow from (61), (22), (27), (30), and (31).


## 3. Proof of Theorem 3

This section presents the proof of Theorem3.Proof of Theorem 3.
Takingφ=uϵ−1ζs as a test function in (2), where 0<ϵ<1, s is sufficiently large, and ζx is as in the proof of Lemma 2, then standard calculations imply:(62)1ϵ∫Bρuϵx,zζsdx−1ϵ∫Bρuϵx,t2ζsdx+s∫t/2z∫Bρum−1p−1+ϵ−1Dup−2DuDζζs−1dxdτ=1−ϵ∫t/2z∫Bρum−1p−1+ϵ−2Dupζsdxdτ+∫t/2z∫Bρaτ−σuq+ϵ−1Dulζsdxdτ,where 0<t/2<z<t<T. Applying Young’s inequality, we obtain(63)∫Bρuϵx,zζsdx−∫Bρuϵx,t2ζsdx≥−γρ−p∫t/2z∫Bρump−1−1+δζs−pdxdτ+a∫t/2z∫Bρτ−σuq+ϵ−1Dulζsdxdτ.
Note that(64)∫t/2z∫Bρτ−σuq+ϵ−1Dulζsdxdτ≥γρ−l∫t/2z∫Bρτ−σuq+l+ϵ−1ζsdxdτ,where we have used Gagliardo–Nirenberg inequality in [18] and Lemma 5.1 in [46] with ρ=ρt=γ0t1/p+θmp−1−1.
Inserting (64) into (63), we obtain(65)∫Bρuϵx,zζsdx−∫Bρuϵx,t2ζsdx≥−γρ−p∫t/2z∫Bρump−1−1+ϵζs−pdxdτ+γρ−l∫t/2z∫Bρτ−σuq+l+ϵ−1ζsdxdτ≥−γρ−pq+l+ϵ−1−lmp−1−1+ϵ/q+l−mp−1+N∫t/2zτσmp−1−1+ϵ/q+l−mp−1dτ+γρ−l∫t/2z∫Bρτ−σuq+l+ϵ−1ζsdxdτ≥−γρ−pq+l+ϵ−1−lmp−1−1+ϵ/q+l−mp−1+N∫t/2zτσmp−1−1+ϵ/q+l−mp−1dτ+γρ−l−Nq+l−1/ϵ∫t/2zτ−σ∫Bρuϵζsdxq+l+ϵ−1/ϵdτ,for all t/2<z<t, where we have used Young’s inequality and q+l≥mp−1 in the second inequality and Hölder’s inequality in the third inequality. Therefore, ∫Bρuϵx,zζsdx majorizes the solution yz to(66)y′=−γρ−pq+l+ϵ−1−lmp−1−1+ϵ/q+l−mp−1+Nzσmp−1−1+ϵ/q+l−mp−1+γρ−l−Nq+l−1/ϵz−σyq+l+ϵ−1/ϵ,int2,t,yt2=∫Bρuϵx,t2ζsdx.
Thus, the solutiony of problem (66) is bounded over t/2,t. It follows from Lemma 4.1 in [17] that(67)∫Bρuϵx,t2ζsdx≤γmaxρϵl/q+l−1+Nt−ϵ1−σ/q+l−1,ρϵl−p/q+l−mp−1+Ntϵσ/q+l−mp−1.
By Lemma 5.1 of [46], we have(68)∫Bρuϵx,t2ζsdx≥γρ−ϵθ+N,for t small and ρ=ρt defined above.
Lett⟶0, it is easily seen that (67) and (68) imply. σ+θpq+l−lmp−1−p+l/pp+θmp−1−1≤p−l/p.


## 4. Conclusion

The local existence of solutions of the doubly degenerate parabolic equation with time-dependent gradient source and initial data measures is studied in this paper. The equation is a class of non-Newtonian polytropic filtration equation and contains the heat equation, the porous medium equation, the evolutionary p-Laplacian equation, and so on. The main difficulties to establish the desired results here are the complicated structure of the equation and the lower regularity of the initial data. By making full use of the structure of the equation and the delicate a priori estimates, we obtain the local existence of weak solutions under the restriction conditionσ+θpq+l−lmp−1−p+l/pp+θmp−1−1<p−l/p. Furthermore, if σ+θpq+l−lmp−1−p+l/pp+θmp−1−1>p−l/p, a counterexample indicates that the equation considered here has no nonnegative solution. We also remark that whether the existence of solutions holds for the limiting case σ+θpq+l−lmp−1−p+l/pp+θmp−1−1=p−l/p is an interesting problem and left open, since the methods used here are invalid for this case.



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