Two Remarks on the Local Behavior of Solutions to Logarithmically Singular Diffusion Equations and its Porous-Medium Type Approximations
aa r X i v : . [ m a t h . A P ] M a y Two Remarks on the Local Behavior of Solutionsto Logarithmically Singular Diffusion Equationsand its Porous-Medium Type Approximations
Emmanuele DiBenedettoDepartment of Mathematics, Vanderbilt University1326 Stevenson Center, Nashville TN 37240, USAemail: [email protected]
Ugo GianazzaDipartimento di Matematica “F. Casorati”, Universit`a di Paviavia Ferrata 1, 27100 Pavia, Italyemail: [email protected]
Naian LiaoDepartment of Mathematics, Vanderbilt University1326 Stevenson Center, Nashville TN 37240, USAemail: [email protected]
Abstract
For the logarithmically singular parabolic equation (1.1) below, weestablish a Harnack type estimate in the L topology, and we show thatthe solutions are locally analytic in the space variables and differentiable intime. The main assumption is that ln u possesses a sufficiently high degreeof integrability (see (1.3) for a precise statement). These two propertiesare known for solutions of singular porous medium type equations (0 0, provided the indicated sufficiently high order of integrabilityis in force. The latter then appears as the discriminating assumptionbetween solutions of parabolic equations with power-like singularities andlogarithmic singularities to insure such solutions to be regular. AMS Subject Classification (2000): Primary 35K65, 35B65; Sec-ondary 35B45 Key Words: Singular parabolic equations, L loc -Harnack estimates, an-alyticity. Main Results We continue here the investigations initiated in [2, 3], on the local behaviorof non-negative solutions to logarithmically singular parabolic equations of thetype u ∈ C loc (cid:0) , T ; L ( E ) (cid:1) , ln u ∈ L (cid:0) , T ; W , ( E ) (cid:1) ; u t − ∆ ln u = 0 weakly in E T = E × (0 , T ] (1.1)where E is an open set in R N and T > 0. It is assumed throughout that u ∈ L r loc ( E T ) for some r > max (cid:8) N (cid:9) (1.2)and that ln u ∈ L ∞ loc (cid:0) , T ; L p loc ( E ) (cid:1) for some p ≥ . (1.3)The modulus of ellipticity of the principal part is u − . Therefore the equationis degenerate as u → ∞ and singular as u → 0. It was shown in [2] that(1.2) implies that u is locally bounded in E T , and hence the equation is notdegenerate. Likewise if (1.3) holds for some p > N + 2, then the solution u islocally bounded below, and hence the equation is not singular. As a consequence u is locally, a classical solution to (1.1). This was realized by establishing a localupper and lower bound on u , via a pointwise Harnack-type estimate.The main results of this note are that if u is a locally bounded, weak solutionsto (1.1), then: i. If ln u satisfies (1.3) for some p ≥ 2, then u satisfies a local Harnack in-equality in the L topology, as opposed to a pointwise Harnack estimate. ii. If ln u satisfies (1.3) for some p > N + 2, then u is locally analytic in thespace variables uniformly in t , and C ∞ in time. L ρ > K ρ be the cube centered at the origin of R N and edge ρ , and for y ∈ R N let K ρ ( y ) denote the homothetic cube centered at y . Moreover, Q ρ ( θ )denotes the parabolic cylinder K ρ × ( − θρ , < s < t ≤ T and y ∈ E let ρ be so small that K ρ ( y ) × ( s, t ] ⊂ E T . Since u ∈ L ∞ loc ( E T ) the quantity M = ess sup K ρ × ( s,t ] u (2.1)is well defined and finite. Also, if (1.3) holds then the quantityΛ p = ess sup s ≤ τ ≤ t (cid:16) − Z K ρ ( y ) (cid:12)(cid:12)(cid:12) ln u ( x, τ ) M (cid:12)(cid:12)(cid:12) p dx (cid:17) p (2.2)is well defined and finite. 2 roposition 2.1 Let u be a non-negative, local, weak solution to (1.1) satis-fying in addition (1.2) and (1.3) for some p ≥ . There exists a positive con-stant γ depending only on { N, r, p } and Λ and Λ , such that for all cylinders K ρ ( y ) × [ s, t ] ⊂ E T , there holds sup s<τ 1) is to be chosen, and q > θ > x o , t o ) + Q ρ ( θ ) = K ρ ( x o ) × ( t o − θ (8 ρ ) , t o ] ⊂ E T . (3.2)These are backward, parabolic cylinders with “vertex“ at ( x o , t o ) whose heightdepends on the solution itself through the quantity θ . In this sense they areintrinsic to the solution itself.Continue to assume that u satisfies (1.2) and let ln u satisfy (1.3) for some p > N + 2. Then M = ess sup ( x o ,t o )+ Q ρ ( θ ) u (3.3)is well defined and finite. Moreover the dimensionless quantity η = h − Z K ρ ( x o ) (cid:16) u ( x, t o ) M (cid:17) q dx i q r − N = (cid:16) θεM (cid:17) r − N (3.4)is well defined and strictly positive. Finally the dimensionless quantityΛ p = ess sup t o − θ (8 ρ ) <τ Let u be a non-negative,local, weak solution to (1.1), satisfying the integrability conditions (1.2) and(1.3) for some p > N + 2 , and assume θ > . There exist a constant ε ∈ (0 , ,and a continuous, increasing function η → f ( η, Λ p ) defined in R + and vanishingat η = 0 , that can be quantitatively determined apriori only in terms of { N, p, q } ,and Λ p , such that inf K ρ ( x o ) u ( · , t ) ≥ f ( η, Λ p ) sup ( x o ,t o )+ Q ρ ( θ ) u for all t ∈ ( t o − θρ , t o ] (3.6) For η → and Λ p → ∞ , the function η → f ( η, Λ p ) can be taken to be of theform f ( η ) = exp n − Λ C p η C o for ≤ η ≪ and Λ p ≫ or positive constants C and C that can be determined apriori only in termsof { N, p, q } . Moreover ε → and C + C → ∞ as p → ∞ or p → N + 2 . (3.8) Remark 3.1 In [2] the constant η was given a more general form. For the pur-pose of this note the definition (3.4) represents the degeneracy of the equation,quantified by M → ∞ . The occurrence Λ p → ∞ quantifies, roughly speaking,the singularity of the equation. ( x o , t o ) Theorem 3.2 Let u be a non-negative, local, weak solution to (1.1), satisfyingthe integrability conditions (1.2) and (1.3) for some p > N + 2 , and assume θ > . There exist two parameters C and H , that have a polynomial dependenceon f ( η ) , [ f ( η )] − , N , such that for every N -dimensional multi-index α | D α u ( x o , t o ) | ≤ CH | α | | α | ! ρ | α | u ( x o , t o ) . (3.9) Moreover, for every non-negative integer k (cid:12)(cid:12)(cid:12) ∂ k ∂t k u ( x o , t o ) (cid:12)(cid:12)(cid:12) ≤ CH k (2 k )! ρ k u ( x o , t o ) − k . (3.10) Remark 3.2 The theorem continues to hold, with the same assumptions, forlocal, weak solutions to the quasilinear equations (2.5), provided the function A is analytic in all its arguments whenever u is bounded above and below bypositive constants. Consider local, non-negative, weak solutions in E T to the porous medium equa-tion u ∈ C loc (cid:0) , T ; L ( E ) (cid:1) , w ∈ L (cid:0) , T ; W , ( E ) (cid:1) ; u t − ∆ w = 0 weakly in E T = E × (0 , T ] (4.1)where w = u m − m for 0 < m ≪ . (4.2)As m → 0, formally (4.1)–(4.2) tend to (1.1). In [3] a precise topology wasintroduced by which such a formal limit is rigorous. A natural question iswhether solutions to (4.1)–(4.2) satisfy a version of the L Harnack estimate(2.3), which as m → .1 Harnack Type Estimates in the Topology of L , forWeak Solutions to (4.1)–(4.2) A first statement in this direction is that u satisfiessup s<τ 1) + 2 . (4.4)Here γ depends upon N and m and γ ( m ) → ∞ as m → 0. Thus, one cannotformally recover (2.3) by letting m → u ∈ C loc (cid:0) , T ; L ( E ) (cid:1) , w ∈ L (cid:0) , T ; W , ( E ) (cid:1) u t − div A ( x, t, u, Du ) = 0 weakly in E T (4.5)where the function A : E T × R N +1 → R N is only assumed to be measurableand subject to the structure conditions A ( x, t, u, p ) · p ≥ C o u m − | p | | A ( x, t, u, p ) | ≤ C u m − | p | a.e. in E T , (4.6)where C o and C are given positive constants. In such a case the constant γ in(4.3) depends also on these structural constants. The proof of these statementsis in [4], Appendix B.A major difference between (2.3) and (4.3) is that in the latter u is notrequired to be locally bounded, nor does γ depend on some analogue of thequantity Λ p as defined in (2.2). This raises the question as to whether (4.3)holds with γ independent of m but dependent on some analogue of Λ p .Henceforth we assume u ∈ L r loc ( E T ) for some r > max (cid:8) N (1 − m ) (cid:9) (4.7)and that w ∈ L ∞ loc (cid:0) , T ; L p loc ( E ) (cid:1) for some p ≥ . (4.8)It was shown in [4] that (4.7) implies that u ∈ L ∞ loc ( E T ) and hence the corre-sponding quantity M defined as in (2.1) is well defined and finite. SetΛ m,p = ess sup s<τ 0, provided proper conditions are placed, that insure thepointwise convergence of the solutions to (4.1)-(4.2) to solutions of (1.1). Theseconditions are identified in [3] and we will touch on them briefly in the nextsubsection. ( x o , t o ) Having fixed ( x o , t o ) ∈ E T and K ρ ( x o ) ⊂ E , the intrinsic geometry of (4.1)–(4.2) is determined by θ m = ε (cid:16) − Z K ρ ( x o ) u q ( x, t o ) dx (cid:17) − mq . (4.10)The intrinsic cylinders are as in (3.2) with θ replaced by θ m . The analogues of η in (3.4) are σ = h − Z K ρ ( x o ) (cid:0) u ( x, t o ) M (cid:17) q dx i q λr (4.11)where r ≥ λ r = N ( m − 1) + 2 r > . (4.12)In [4] a Harnack estimate of the form of (3.6) was proved for these solutionswith f ( · ) depending only on σ and of the form f ( σ ) = σ β γ ( m ) (4.13)where γ ( m ) → ∞ as m → 0. The constant β depends on λ r and β ( λ r ) → ∞ as λ r → 0. It was observed in [4] § m ∈ (0 , x o , t o ), and at least the Lipschitz continuity in time.Because of the indicated dependence of γ ( m ) on m in (4.13), such a regularitydoes not directly carry to the limit as m → 0. In [3] we established a Harnackestimate of the form (3.6) for solutions to (4.1)–(4.2) and its quasi-linear versions(4.5)-(4.6), with f ( · ) depending on σ , as defined in (4.11), and Λ m,p as definedin (4.9) provided p > N + 2. The form of such f ( · ) is the same as that in (3.7)with the proper change in symbolism. The new feature of such an f ( · ) is that,while depending on the quantities σ and Λ m,p , each quantifying the degeneracyand the singularity of the equation, is independent of m and hence is “stable”as m → 0, provided σ and Λ m,p are uniformly estimated with respect to m .7s a consequence, the analyticity estimates of Theorem 3.2, can be recoveredfrom the analogous ones for solutions to (4.1)–(4.2) whenever solutions { u m } to the latter converge pointwise to solutions to the former. In [3] it was shownthat this occurs if there exists m ∗∗ ∈ (0 , 1) such that u m ∈ L ∞ loc (cid:0) , T ; L r loc ( E ) (cid:1) for some r > max { N } w m ∈ L ∞ loc (0 , T ; L p loc ( E ) (cid:1) for some p > N + 2 (4.14)uniformly in m ∈ (0 , m ∗∗ ). It is also required that there exists an open set E o ⊂ E and a positive number σ E o ; T such that Z E o u m ( · , T ) dx ≥ σ E o ; T uniformly in m. (4.15) The proof is a local version of an argument of [1] for global solutions to theporous medium equation for 0 < m < 1. Let ζ ∈ C ∞ o ( R N ) be such that ≤ ζ ≤ ζ = 1 in K ρ ζ = 0 in R N \ K ρ . (5.1)Then, by the divergence theorem Z R N ∆ ζ dx = Z K ρ ∆ ζ dx = Z ∂K ρ ∂ζ ∂n ds = 0 . (5.2)By (5.2), for any positive constant M , any ζ as in (5.1), and any non-negativefunction v such that ∆ ζ ln v is integrable, we have Z K ρ ∆ ζ ln vdx = Z K ρ ∆ ζ ln( vM ) dx. (5.3)Now consider ζ ∈ C ∞ o (0 , + ∞ ) and ζ as in (5.1), such that | Dζ | ≤ C ( N ) ρ , | ∆ ζ | ≤ C ( N ) ρ . By the previous notation, with u a solution to (1.1), we have − Z ∞ Z R N ζ ′ ζ udxdt = Z ∞ Z R N ζ ∆ ζ ln udxdt. Taking into account (5.3), for any positive constant Mddt Z K ρ ζ udx = Z K ρ ∆ ζ ln( uM ) dx in D ′ (0 , T ) , L loc (0 , T ). Therefore (cid:12)(cid:12)(cid:12) ddt Z K ρ ζ udx (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z K ρ ∆ ζ ln( uM ) dx (cid:12)(cid:12)(cid:12) ≤ Z K ρ | ∆ ζ | (cid:12)(cid:12)(cid:12) ln( uM ) (cid:12)(cid:12)(cid:12) dx. By the definition of Λ p and from the previous estimate, (cid:12)(cid:12)(cid:12) ddt Z K ρ ζ udx (cid:12)(cid:12)(cid:12) ≤ Λ p Z K ρ | ∆ ζ | p ′ dx ! p ′ | K ρ | p ≤ Λ p C ( N ) ρ ρ Np ′ ρ Np = C (Λ p , N ) ρ λ , where λ = 2 − N . Taking into account the size of the support of ζ , for any0 < s < t < T we conclude Z K ρ u ( x, t ) dx ≤ C (Λ p , N ) Z K ρ u ( x, s ) dx + t − sρ λ ! . Lemma 6.1 Let u be a non-negative, local, weak solution to the quasi-linearsingular equations (2.5)–(2.6), in E T . There exist two positive constants γ , γ depending only on the data { N, C o , C } , such that for all cylinders K ρ ( y ) × [ s, t ] ⊂ E T , and all σ ∈ (0 , , Z ts Z K ρ ( y ) | Du | u ζ dxdτ ≤ γ (1 + Λ ) S σ + γ σ (Λ + Λ ) (cid:18) t − sρ λ (cid:19) , where S σ = sup s<τ Assume ( y, s ) = (0 , σ ∈ (0 , x → ζ ( x ) be a non-negative piecewise smooth cutoff function in K (1+ σ ) ρ that vanishes outside K (1+ σ ) ρ , equals one on K ρ , and such that | Dζ | ≤ ( σρ ) − . Let s ∈ [0 , t ] besuch that S σ = sup 9n the weak formulation of (2.5)–(2.6), take the test function ϕ = (cid:18) ln ¯ S σ u (cid:19) + ζ and integrate over Q = K (1+ σ ) ρ × (0 , t ], to obtain0 = Z Z Q ∂∂τ u (cid:18) ln ¯ S σ u (cid:19) + ζ dx dτ + Z Z Q A ( x, τ, u, Du ) · D "(cid:18) ln ¯ S σ u (cid:19) + ζ dx dτ = I + I . We estimate these two terms separately. I = Z Z Q ∂∂τ u (cid:18) ln ¯ S σ u (cid:19) + ζ dx dτ = Z Z Q ∩ [ u< ¯ S σ ] ∂∂τ u (cid:18) ln ¯ S σ u (cid:19) ζ dx dτ = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) (cid:18) u ln ¯ S σ u + u (cid:19) ( x, t ) dx − Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) (cid:18) u ln ¯ S σ u + u (cid:19) ( x, dx. Next, I = Z Z Q A ( x, τ, u, Du ) · D "(cid:18) ln ¯ S σ u (cid:19) + ζ dx dτ = Z Z Q ∩ [ u< ¯ S σ ] A ( x, τ, u, Du ) · D (cid:20)(cid:18) ln ¯ S σ u (cid:19) ζ (cid:21) dx dτ = − Z Z Q ∩ [ u< ¯ S σ ] ζ A ( x, τ, u, Du ) Duu dx dτ + 2 Z Z Q ∩ [ u< ¯ S σ ] ζ ln ¯ S σ u A ( x, τ, u, Du ) Dζdx dτ ≤ − C o Z Z Q ∩ [ u< ¯ S σ ] ζ | Du | u dx dτ + 2 C Z Z Q ∩ [ u< ¯ S σ ] ζ ln ¯ S σ u | Du | u | Dζ | dx dτ ≤ − C o Z Z Q ∩ [ u< ¯ S σ ] ζ | Du | u dx dτ + γσ ρ Z Z Q ∩ [ u< ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln ¯ S σ u (cid:12)(cid:12)(cid:12)(cid:12) dx dτ, where γ = 2 C C o . Therefore, we conclude that C o Z Z Q ∩ [ u< ¯ S σ ] ζ | Du | u dx dτ ≤ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) (cid:18) u ln ¯ S σ u + u (cid:19) ( x, t ) dx γσ ρ Z Z Q ∩ [ u< ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln ¯ S σ u (cid:12)(cid:12)(cid:12)(cid:12) dx dτ = J + J . We have J = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) (cid:18) u ln ¯ S σ u + u (cid:19) ( x, t ) dx ≤ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] (cid:18) u ln ¯ S σ u + u (cid:19) ( x, t ) dx = ¯ S σ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] u ¯ S σ ln ¯ S σ u ( x, t ) dx + Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] u ( x, t ) dx ≤ ¯ S σ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ln ¯ S σ u ( x, t ) dx + S σ ≤ γ S σ − Z K (1+ σ ) ρ ln Mu ( x, t ) dx + S σ ≤ γ Λ S σ + S σ = γ (1 + Λ ) S σ , where γ = 2 N . Moreover, J = γσ ρ Z Z Q ∩ [ u< ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln ¯ S σ u (cid:12)(cid:12)(cid:12)(cid:12) dx dτ ≤ γσ ρ Z Z Q ∩ [ u< ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln Mu (cid:12)(cid:12)(cid:12)(cid:12) dx dτ ≤ γσ ρ Z Z Q (cid:12)(cid:12)(cid:12)(cid:12) ln Mu (cid:12)(cid:12)(cid:12)(cid:12) dx dτ = γσ ρ Z t Z K (1+ σ ) ρ (cid:12)(cid:12)(cid:12)(cid:12) ln Mu (cid:12)(cid:12)(cid:12)(cid:12) dx dτ ≤ γσ ρ sup <τ Z Z Q ∩ [ u< ¯ S σ ] ζ | Du | u dx dτ ≤ γ (Λ + 1) S σ + γσ Λ (cid:18) tρ λ (cid:19) . (6.1)Now, if we take the test function ϕ = (cid:18) ln u ¯ S σ (cid:19) + ζ in the weak formulation of (2.5)–(2.6) and integrate over Q = K (1+ σ ) ρ × (0 , t ],we obtain0 = Z Z Q ∂∂τ u (cid:18) ln u ¯ S σ (cid:19) + ζ dx dτ + Z Z Q A ( x, τ, u, Du ) · D "(cid:18) ln u ¯ S σ (cid:19) + ζ dx dτ = I + I . 11e estimate these two terms separately. I = Z Z Q ∂∂τ u (cid:18) ln u ¯ S σ (cid:19) + ζ dx dτ = Z Z Q ∩ [ u> ¯ S σ ] ∂∂τ u (cid:18) ln u ¯ S σ (cid:19) ζ dx dτ = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) (cid:18) u ln u ¯ S σ − u (cid:19) ( x, t ) dx − Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) (cid:18) u ln u ¯ S σ − u (cid:19) ( x, dx. Next, I = Z Z Q A ( x, τ, u, Du ) · D "(cid:18) ln u ¯ S σ (cid:19) + ζ dx dτ = Z Z Q ∩ [ u> ¯ S σ ] A ( x, τ, u, Du ) · D (cid:20)(cid:18) ln u ¯ S σ (cid:19) ζ (cid:21) dx dτ = Z Z Q ∩ [ u> ¯ S σ ] ζ A ( x, τ, u, Du ) Duu dx dτ + 2 Z Z Q ∩ [ u> ¯ S σ ] ζ ln u ¯ S σ A ( x, τ, u, Du ) Dζdx dτ ≥ C o Z Z Q ∩ [ u> ¯ S σ ] ζ | Du | u dx dτ − C Z Z Q ∩ [ u> ¯ S σ ] ζ ln u ¯ S σ | Du | u | Dζ | dx dτ ≥ C o Z Z Q ∩ [ u> ¯ S σ ] ζ | Du | u dx dτ − γσ ρ Z Z Q ∩ [ u> ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln u ¯ S σ (cid:12)(cid:12)(cid:12)(cid:12) dx dτ, where again γ = 2 C C o . Therefore, we conclude that C o Z Z Q ∩ [ u> ¯ S σ ] ζ | Du | u dx dτ ≤ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) (cid:18) u ln u ¯ S σ (cid:19) ( x, dx + Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) u ( x, t ) dx + γσ ρ Z Z Q ∩ [ u> ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln u ¯ S σ (cid:12)(cid:12)(cid:12)(cid:12) dx dτ ≤ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) (cid:18) u ln u ¯ S σ (cid:19) ( x, dx + S σ + γσ ρ Z Z Q ∩ [ u> ¯ S σ ] (cid:12)(cid:12)(cid:12)(cid:12) ln u ¯ S σ (cid:12)(cid:12)(cid:12)(cid:12) dx dτ ≤ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) (cid:18) u ln M ¯ S σ (cid:19) ( x, dx + S σ + γσ ρ (cid:12)(cid:12)(cid:12)(cid:12) ln M ¯ S σ (cid:12)(cid:12)(cid:12)(cid:12) tρ N ≤ S σ + (cid:18) ln M ¯ S σ (cid:19) S σ + γσ (cid:12)(cid:12)(cid:12)(cid:12) ln M ¯ S σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) tρ λ (cid:19) . 12e need to evaluate ln M/ ¯ S σ . As in the interval (0 , 1] the function f ( s ) = − ln s is convex, Jensen’s inequality yieldsln M ¯ S σ = ln M − R K (1+ σ ) ρ u ( x, s ) dx = − ln − Z K (1+ σ ) ρ u ( x, s ) M dx ≤ − Z K (1+ σ ) ρ − ln u ( x, s ) M dx = − Z K (1+ σ ) ρ ln Mu ( x, s ) dx ≤ γ Λ , where γ = 2 N . Hence, we have Z Z Q ∩ [ u> ¯ S σ ] ζ | Du | u dx dτ ≤ γ (Λ + 1) S σ + γσ Λ (cid:18) tρ λ (cid:19) . (6.2)The lemma follows by combining estimates (6.1) and (6.2).The use of (cid:16) ln ¯ S σ u (cid:17) + ζ as test function can be justified using (cid:16) ln ¯ S σ u + ǫ (cid:17) + andthen letting ǫ → Corollary 6.1 Let u be a non-negative, local, weak solution to the singularequations (2.5)–(2.6), in E T . There exists a positive constant γ depending onlyon the data { N, C o , C } , such that for all cylinders K ρ ( y ) × [ s, t ] ⊂ E T , and all σ ∈ (0 , , ρ Z ts Z K ρ ( y ) | A ( x, τ, u, Du ) | dx dτ ≤ γ max n (1 + Λ ) ; (Λ + Λ ) o (cid:20) S σ + 1 σ (cid:16) t − sρ λ (cid:17)(cid:21) (cid:18) t − sρ λ (cid:19) . Proof - Assume ( y, s ) = (0 , Q = K ρ × (0 , t ]. By the structureconditions of A ρ Z t Z K ρ | A ( x, τ, u, Du ) | dx dτ ≤ C ρ Z Z Q | Du | u dx dτ ≤ C ρ (cid:18)Z Z Q | Du | u dx dτ (cid:19) ρ N t = C (cid:18)Z Z Q | Du | u dx dτ (cid:19) (cid:18) tρ λ (cid:19) . By Lemma 6.1 we conclude. Assume ( y, s ) = (0 , n = 0 , , . . . set ρ n = n P j =1 j ρ, K n = K ρ n ; ˜ ρ n = ρ n + ρ n +1 , ˜ K n = K ˜ ρ n x → ζ n ( x ) be a non-negative, piecewise smooth cutoff function in ˜ K n that equals one on K n , and such that | Dζ n | ≤ n +2 /ρ . In the weak formulationof (2.5)–(2.6) take ζ n as a test function, to obtain Z ˜ K n u ( x, τ ) ζ n dx ≤ Z ˜ K n u ( x, τ ) ζ n dx + 2 n +2 ρ (cid:12)(cid:12)(cid:12) Z τ τ Z ˜ K n | A ( x, τ, u, Du ) | dx dτ (cid:12)(cid:12)(cid:12) ≤ Z ˜ K n u ( x, τ ) ζ n dx + γ n S n +1 (cid:16) tρ λ (cid:17) + γ n (cid:16) tρ λ (cid:17) , where S n = sup ≤ τ ≤ t Z K n u ( · , τ ) dx. Since the time levels τ and τ are arbitrary, choose τ one for which Z K ρ u ( · , τ ) dx = inf ≤ τ ≤ t Z K ρ u ( · , τ ) dx def = I . With this notation, the previous inequality takes the form S n ≤ I + γ (cid:0) data , Λ , Λ (cid:1) n (cid:16) tρ λ (cid:17) + γ (cid:0) data , Λ , Λ (cid:1) n S n +1 (cid:16) tρ λ (cid:17) . By Young’s inequality, for all ε o ∈ (0 , S n ≤ ε o S n +1 + γ (cid:0) data , Λ , Λ , ε o (cid:1) n h I + (cid:16) tρ λ (cid:17)i . From this, by iteration S o ≤ ε no S n + γ (data , Λ , Λ , ε o ) h I + (cid:16) tρ λ (cid:17)i n − P i =0 (4 ε o ) i . Choose ε o so that the last term is majorized by a convergent series, and let n → ∞ .The proof of Proposition 4.1 for weak solutions to the porous medium typequasilinear equations (4.5)–(4.6), is similar, with the obvious modifications, andwe confine it to Appendix B. Let u be a non-negative, local, weak solution to (1.1), satisfying the integrabilityconditions (1.2) and (1.3) for some p > N + 2.Fix ( x o , t o ) ∈ E T , assume that K ρ ( x o ) ⊂ E , and assume that the quantity θ defined in (3.1) is positive. The cylinder ( x o , t o ) + Q ρ ( θ ) is assumed to becontained in the domain of definition of u as in (3.2). The quantities M , η andΛ p are defined as in (3.3)–(3.5). 14rom the Harnack-type inequality (3.6),[ f ( η )] u ( x o , t o ) ≤ u ( x, t ) ≤ [ f ( η )] − u ( x o , t o ) (7.1)for any ( x, t ) within the cylinder Q ≡ K ρ ( x o ) × ( t o − θρ , t o ] . (7.2)Equation (1.1) can be rewritten as u t − div( 1 u Du ) = 0 . (7.3)By (7.1) this can be regarded as a particular instance of a linear parabolic equa-tion with bounded and measurable coefficients. By known results (for example,[7], Chapter II) local, weak solutions to (7.3) are locally bounded and locallyH¨older continuous. Consequently, (7.3) can be regarded as a linear parabolicequation with bounded, and H¨older continuous coefficients. Again by classicaltheory (see [7], Chapter III), one can conclude that local, weak solutions areindeed C ∞ with respect to the space variable.By (7.1) the quantity θ can be estimated as θ ≤ ε sup K ρ ( x o ) u ( · , t o ) ≤ ε [ f ( η )] − u ( x o , t o ) θ ≥ ε inf K ρ ( x o ) u ( · , t o ) ≥ ε [ f ( η )] u ( x o , t o ) . (7.4)Let δ = δ ( η ) def = ε [ f ( η )] and introduce the change of variables x → x − x o ρ , t → t − t o u ( x o , t o ) ρ , v = uu ( x o , t o ) . It maps Q onto to˜ Q def = K × ( − θu ( x o , t o ) , ⊃ Q δ def = K × ( − δ , , (7.5)and within Q δ the function v satisfies v t − v ∆ v = − | Dv | v , (7.6)with f ( η ) ≤ v ≤ f ( η ) − . By a result of [6], there exist constants 0 < σ < C and H such thatsup Q σδ | D α v | ≤ CH | α | | α | ! , sup Q σδ (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂t k v (cid:12)(cid:12)(cid:12)(cid:12) ≤ CH k (2 k )! (7.7)15here Q σδ = K σ × ( − σδ, C = γ C o , H = γ max { C o [ f ( η )] − , [ f ( η )] − } (7.8)where γ and γ are constants independent of v and C o is a function of f ( η )and satisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂t k D α v (cid:12)(cid:12)(cid:12)(cid:12) ≤ C o in Q δ for | α | + 2 k ≤ N , where [ a ] denotes the integer part of a . Thus in particular an upper bound onthese derivatives up to the indicated order, gives their analyticity as signified by(7.7). Assuming such an upper bound for the moment, we return to the originalcoordinates to get | D α u ( x o , t o ) | = | D α v (0 , | u ( x o , t o ) ρ | α | ≤ CH | α | | α | ! ρ | α | u ( x o , t o ) , (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂t k u ( x o , t o ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂t k v (0 , (cid:12)(cid:12)(cid:12)(cid:12) u ( x o , t o ) − k ρ k ≤ CH k (2 k )! ρ k u ( x o , t o ) − k . (7.9)The proof is concluded, once the dependence of C o on f ( η ) is determined. Thisestimation can be achieved by local DeGiorgi’s or Moser’s estimates. While themethod is known, it is technically involved and reported in detail in Appendix A.The analogous analyticity estimates for solutions to the porous medium typeequation (4.1)–(4.2) are similar, with the obvious changes, and we omit thedetails. Appendices A Analyticity in the Space Variables, of Solu-tions to (1.1). Estimating the first N ] + 16 Derivatives of v . We will use expressions such as w f ( w ), w f ′ and similar ones, but we only haveat our disposal the notion of weak solution, and therefore, such a way of workingdoes not seem justified. However, by the Harnack estimate of Theorem 3.1,solutions are classical, and in these calculations we are turning the qualitative information of u being classical into the quantitative information of u beinganalytic.With respect to the previous sections, we use a different notation for cylin-ders, and we let Q ( ρ, θ ) = K ρ × ( − θ, .1 An estimate of k Dv k ∞ Take D x i of the logarithmic diffusion equation and set w i = v x i to get ∂ t w i − div (cid:0) v Dw i − v w i Dv (cid:1) = 0Setting w = ( w , . . . , w N ), yields w t − div (cid:0) v Dw − v w ⊗ w (cid:1) = 0 . (A.1)For all derivations below we stipulate that λ < 1, Λ = λ − > λ < v < Λ, θ < ρ < Proposition A.1 Let w be a solution to (A.1) and ζ be a cutoff function in Q = Q ( ρ, θ ) . Then sup − θ There exists a positive parameter δ that depends only on Λ λ ,such that if ρ ∈ (0 , δ ] then k w k ∞ ,Q ( σρ,σθ ) ≤ γ (cid:18) Λ λ (cid:19) µ (1 + θ − µ )(1 − σ ) µ . where γ , µ , and µ are positive parameters that depend only on N . roof - We first estimate k w k . In the weak formulation of (1.1) take the testfunction vζ . By standard calculations λ Z Z Q ( ρ,θ ) | Dv | ζ dxdt ≤ γ Z K ρ ×{− θ } v ζ dx + γ Z Z Q ( ρ,θ ) v | Dζ | dxdt. Hence Z Z Q ( σρ,σθ ) | Dv | dxdt ≤ γ Λ λ ρ N (1 − σ ) (cid:20) θρ (cid:21) i ≤ γ (cid:18) Λ λ (cid:19) θ (1 − σ ) ρ , (A.2)The energy estimate (A.1) with f ( | w | ) = | w | β and β ≥ 0, yieldssup − θ Let v be a classical solution to the logarithmic diffusion equa-tion and assume < λ ≤ v − ≤ Λ in Q ( ρ, θ ) ; then k v t k ∞ ,Q ( σρ,σθ ) ≤ γ (cid:18) Λ λ (cid:19) µ (1 + θ − µ )(1 − σ ) µ , where γ , µ , and µ are positive parameters that depend only on N .Proof - Multiply (1.1) by the test function v t ζ and integrate over the cylin-der Q = Q ( ρ + σρ , θ + σθ ), where σ ∈ (0 , ζ vanishes on the parabolicboundary of Q and takes value 1 in Q ( σρ, σθ ). A standard calculation gives0 = Z Z Q (cid:20) v t ζ dxdt + 1 v Dv [ ζ Dv t + 2 v t ζDζ ] (cid:21) dxdt = Z Z Q (cid:20) v t ζ dxdt + 12 v ζ ∂∂t | Dv | + 2 v v t ζDvDζ (cid:21) dxdt = Z Z Q v t ζ dxdt + Z K ρ + σρ ×{ } v | Dv | ζ dx − Z Z Q | Dv | (cid:20) − v v t ζ + 1 v ζζ t (cid:21) dxdt + Z Z Q v v t ζDvDζdxdt. This gives the estimate Z Z Q v t ζ dxdt ≤ Z Z Q | Dv | v | v t | ζ dxdt + Z Z Q v ζ | ζ t || Dv | dxdt + Z Z Q v | Dv || v t | ζ | Dζ | dxdt ≤ Z Z Q v t ζ dxdt + Z Z Q | Dv | v ζ dxdt + Z Z Q | Dv | v | Dζ | dxdt + Z Z Q | Dv | v | ζ t | dxdt. k Dv k ∞ ,Q of the previous section, we have k v t k ,Q ( σρ,σθ ) ≤ γ (cid:18) Λ λ (cid:19) µ (1 + θ − µ )(1 − σ ) µ . for some µ ( N ), and µ ( N ) > v t f ( | v t | ) ζ where f : R + → R + is a bounded, non-decreasing Lipschitz function, and ζ vanishes onthe parabolic boundary of Q = Q ( ρ, θ ) and takes value 1 in Q ( σρ, σθ ). Let M = k Dv k ∞ ,Q . A standard calculation yields Z Z Q ∂∂t | v t | f ζ dxdt + λ Z Z Q | Dv t | f ζ dxdt + λ Z Z Q | Dv t | | v t | f ′ ζ dxdt ≤ ZZ Q | Dv t || v t | f ζ | Dζ | dxdt + Λ Z Z Q | v t || Dv | (cid:2) | Dv t | f ζ + | v t | f ′ | Dv t | ζ + 2 | v t | f ζ | Dζ | (cid:3) dxdt = I + I + I + I . Let us estimate the four terms. I ≤ λ Z Z Q | Dv t | f ζ dxdt + 16Λ λ Z Z Q | v t | f | Dζ | dxdt ; I ≤ λ Z Z Q | Dv t | f ζ dxdt + 4 M Λ λ Z Z Q | v t | f ζ dxdt ; I ≤ λ Z Z Q | Dv t | | v t | f ′ ζ dxdt + 4 M Λ λ Z Z Q | v t | f ′ ζ dxdt ; I ≤ M Λ Z Z Q | v t | f ζ dxdt + M Λ Z Z Q | v t | f | Dζ | dxdt. Summarizing we havesup − θ 0; then1 β + 2 sup − θ Proposition A.4 Let v be a classical solution to the logarithmic diffusion equa-tion in Q ( ρ, θ ) and fix σ ∈ (0 , . Assume < λ ≤ v − ≤ Λ in Q ( ρ, θ ) . Thereexists a positive parameter δ that depends only Λ λ , such that if ρ ∈ (0 , δ ] , thenin Q ( σρ, σθ ) k w k ∞ ,Q ( σρ,σθ ) ≤ γ (cid:18) Λ λ (cid:19) µ (1 + θ − µ )(1 − σ ) µ , (A.7) where γ , µ and µ are positive parameters that depend only on N and n .Proof - Multiply (A.6) by the test function ∂ k ∂t k D α vf ( | w | ) ζ , where f : R + → R + is a bounded, non decreasing Lipschitz function. Here ζ vanishes on theparabolic boundary of Q = Q ( (1+ σ ) ρ , (1+ σ ) θ ) and takes value 1 in Q ( σρ, σθ ).Standard calculations and a sum over k + | α | = n give Z Z Q ζ ∂∂t Z | w | sf ( s ) ds dxdt + λ Z Z Q | w || D | w || f ′ ( | w | ) ζ dxdt + λ X k + | α | = n Z Z Q | D ∂ k ∂t k D α v | f ( | w | ) ζ dxdt ≤ I, where I = − X k + | α | = n Z Z Q v ζf ( | w | ) ∂ k ∂t k D α v D ∂ k ∂t k D α vDζdxdt − X k + | α | = n Z Z Q (cid:20) X j 0; then the energy estimate yields1 β + 2 sup − θ + σθ 1. We can rewrite (A.6) as ∂ k − ∂t k − D α v t − div (cid:18) v D ∂ k − ∂t k − D α v (cid:19) = div f where f = X j 1. Take the test function ζ D α v , where ζ vanishes on the parabolic boundary of Q and takes 1 in Q ( σρ, σθ ). Z Z Q v | DD α v | ζ dxdt = − ZZ Q ∂∂t ( D α v ) ζ dxdt − Z Z Q v DD α vD α v ζDζdxdt − X | β | < | α | Z Z Q (cid:18) αβ (cid:19) D α − β v DD β vDD α vζ dxdt − X | β | < | α | Z Z Q (cid:18) αβ (cid:19) D α − β v DD β vDD α vD α ζDζdxdt ≤ λ Z Z Q | DD α v | ζ dxdt + (cid:20) − σ ) θ + 1(1 − σ ) ρ (cid:21) P Summing over all | α | = n − X | α | = n Z Z Q | D α v | ζ dxdt ≤ (cid:20) − σ ) θ + 1(1 − σ ) ρ (cid:21) P. If we take into consideration an intermediate cylinder, then this, together with(A.10) in (A.8), yields k w k ∞ ,Q ( σρ,σθ ) ≤ P γ (cid:20) − σ ) θ + 1(1 − σ ) ρ (cid:21) γ (A.11)for some γ depending only on N . The induction hypothesis and the definitionof ρ in (A.5) imply that k w k ∞ ,Q ( σρ,σθ ) ≤ γ ( N, n ) (cid:18) Λ λ (cid:19) µ θ − µ (1 − σ ) µ . (A.12) B Proof of Proposition 4.1 for Weak Solutionsto Equations (4.5)–(4.6) An Auxiliary Lemma Lemma B.1 Let u be a non-negative, local, weak solution to the singular equa-tions (4.5)–(4.6), in E T . There exist two positive constants γ , γ depending nly on the data { N, C o , C } , such that for all cylinders K ρ ( y ) × [ s, t ] ⊂ E T ,and all σ ∈ (0 , , Z ts Z K ρ ( y ) | Du | u − m ζ dx dτ ≤ γ (1 + Λ m , ) ρ N m S − m σ + γ σ ρ (Λ m , + Λ m , ) S m σ ( t − s ) ρ N (1 − m ) , where S σ = sup s<τ In the following we restrict to 0 < m < , since we are mainly interestedin proving the stability of the estimates as m → + . For m ∈ ( , 1) similararguments hold, provided a slightly different test function ϕ is chosen (see [4], § B.1.1 for more details).Assume ( y, s ) = (0 , σ ∈ (0 , x → ζ ( x ) be a non-negativepiecewise smooth cutoff function in K (1+ σ ) ρ that vanishes outside K (1+ σ ) ρ ,equals one on K ρ , and such that | Dζ | ≤ ( σρ ) − . Let s ∈ [0 , t ] be such that S σ = sup 30e estimate these two terms separately. I = Z Z Q ∂∂τ u u − m − ¯ S − m σ m ! + ζ dx dτ = Z Z Q ∩ [ u< ¯ S σ ] ∂∂τ u u − m − ¯ S − m σ m ! ζ dx dτ = − Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) Z ¯ S σ u ( x,t ) s − m − ¯ S − m σ m ds ! dx + Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) Z ¯ S σ u ( x, s − m − ¯ S − m σ m ds ! dx. Next, I = Z Z Q A ( x, τ, u, Du ) · D " u − m − ¯ S − m σ m ! + ζ dx dτ = Z Z Q ∩ [ u< ¯ S σ ] A ( x, τ, u, Du ) · D " u − m − ¯ S − m σ m ! ζ dx dτ = − Z Z Q ∩ [ u< ¯ S σ ] ζ u − m − A ( x, τ, u, Du ) · Du dx dτ + 2 Z Z Q ∩ [ u< ¯ S σ ] ζ u − m − ¯ S − m σ m ! A ( x, τ, u, Du ) · Dζ dx dτ ≤ − C o Z Z Q ∩ [ u< ¯ S σ ] u − m − u m − ζ | Du | dx dτ + 2 C Z Z Q ∩ [ u< ¯ S σ ] ζ u − m − ¯ S − m σ m ! u m − | Du || Dζ | dx dτ ≤ − C o Z Z Q ∩ [ u< ¯ S σ ] ζ u m − | Du | dx dτ + γσ ρ Z Z Q ∩ [ u< ¯ S σ ] u m u − m − ¯ S − m σ m ! dx dτ, where γ = 4 C C o . Therefore, we conclude that C o Z Z Q ∩ [ u< ¯ S σ ] ζ u m − | Du | dx dτ ≤ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) Z ¯ S σ u ( x, s − m − ¯ S − m σ m ds ! dx + γσ ρ Z Z Q ∩ [ u< ¯ S σ ] u m u − m − ¯ S − m σ m ! dx dτ J + J . We have J = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ζ ( x ) Z ¯ S σ u ( x, s − m − ¯ S − m σ m ds ! dx ≤ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ Z ¯ S σ u ( x, (cid:16) ¯ S σ s (cid:17) m − m d (cid:18) s ¯ S σ (cid:19) dx = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ Z (cid:16) u ( x, S σ (cid:17) m y − − m y m − m dy ! dx = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ m (cid:20) m − m y m − − my m (cid:21) (cid:16) u ( x, S σ (cid:17) m dx = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ m " − m − (cid:18) u ( x, S σ (cid:19) − m ! − (cid:18) − u ( x, S σ (cid:19) dx ≤ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ m " m − m − − m u ( x, S σ (cid:18) u ( x, S σ (cid:19) − m − ! dx = Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ − m + 22 − m u ( x, S σ − (cid:16) u ( x, S σ (cid:17) − m m dx ≤ − m Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ dx + 22 − m Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] u ( x, u ( x, − m − ¯ S − m σ m dx = J ′ + J ′′ .J ′ = 12 − m ¯ S − m σ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] dx ≤ γ − m S − m σ ρ N m , where γ = 2 N .J ′′ = 22 − m Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] u ( x, u ( x, − m − ¯ S − m σ m dx = 22 − m Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] ¯ S − m σ u ( x, − m ¯ S − m σ ¯ S m σ − u ( x, m m ¯ S m σ dx ≤ − m ¯ S − m σ Z K (1+ σ ) ρ ∩ [ u< ¯ S σ ] M m − u ( x, m mM m dx ≤ γ − m ¯ S − m σ Λ m , ρ N = 2 γ − m Λ m , S − m σ ρ N m , where γ = 2 N . J ≤ γ (1 + Λ m , ) ρ N m S − m σ . Moreover, J = γσ ρ Z Z Q ∩ [ u< ¯ S σ ] u m u − m − ¯ S − m σ m ! dx dτ = γσ ρ Z Z Q ∩ [ u< ¯ S σ ] u m ¯ S m σ − u m mu m ¯ S m σ ! dx dτ ≤ γσ ρ Z Z Q ∩ [ u< ¯ S σ ] u m (cid:18) M m − u m mM m (cid:19) dx dτ ≤ γσ ρ ¯ S m σ Z Z Q ∩ [ u< ¯ S σ ] (cid:18) M m − u m mM m (cid:19) dx dτ ≤ γσ ρ ¯ S m σ tρ N sup <τ 33e estimate these two terms separately. I = Z Z Q ∂∂τ u ¯ S − m σ − u − m m ! + ζ dx dτ = Z Z Q ∩ [ u> ¯ S σ ] ∂∂τ u ¯ S − m σ − u − m m ! ζ dx dτ = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) Z u ( x,t )¯ S σ ¯ S − m σ − s − m m ds ! dx − Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) Z u ( x, S σ ¯ S − m σ − s − m m ds ! dx. Next, I = Z Z Q A ( x, τ, u, Du ) · D " ¯ S − m σ − u − m m ! + ζ dx dτ = Z Z Q ∩ [ u> ¯ S σ ] A ( x, τ, u, Du ) · D " ¯ S − m σ − u − m m ! ζ dx dτ = 12 Z Z Q ∩ [ u> ¯ S σ ] ζ u − m − A ( x, τ, u, Du ) · Du dx dτ + 2 ZZ Q ∩ [ u> ¯ S σ ] ζ ¯ S − m σ − u − m m ! A ( x, τ, u, Du ) · Dζdx dτ ≥ C o Z Z Q ∩ [ u> ¯ S σ ] ζ u − m − u m − | Du | dx dτ − C Z Z Q ∩ [ u> ¯ S σ ] ζ ¯ S − m σ − u − m m ! u m − | Du || Dζ | dx dτ ≥ C o Z Z Q ∩ [ u> ¯ S σ ] ζ u m − | Du | dx dτ − γσ ρ Z Z Q ∩ [ u> ¯ S σ ] u m ¯ S − m σ − u − m m ! dx dτ, where again γ = 4 C C o . Therefore, we conclude that C o Z Z Q ∩ [ u> ¯ S σ ] ζ u m − | Du | dx dτ ≤ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) Z u ( x, S σ ¯ S − m σ − s − m m ds ! dx + γσ ρ Z Z Q ∩ [ u> ¯ S σ ] u m u m − ¯ S m σ mu m ¯ S m σ ! dx dτ = J + J . 34e have J = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ζ ( x ) Z u ( x, S σ ¯ S − m σ − s − m m ds ! dx ≤ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ Z u ( x, S σ − (cid:16) ¯ S σ s (cid:17) m m d (cid:18) s ¯ S σ (cid:19) dx = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ Z (cid:16) u ( x, S σ (cid:17) m − y − m y m − m dy dx = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ m (cid:20) my m − m − m y m − (cid:21) (cid:16) u ( x, S σ (cid:17) m dx = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ m "(cid:18) u ( x, S σ − (cid:19) − − m (cid:18) u ( x, S σ (cid:19) − m − ! dx ≤ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ m " m − m + 22 − m u ( x, S σ − (cid:18) u ( x, S σ (cid:19) − m ! dx = Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ − m + 22 − m u ( x, S σ − (cid:16) u ( x, S σ (cid:17) − m m dx ≤ − m Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] ¯ S − m σ dx + 22 − m Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] u ( x, 0) ¯ S − m σ − u ( x, − m m dx = J ′ + J ′′ .J ′ = 12 − m ¯ S − m σ Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] dx ≤ γ − m S − m σ ρ N m where γ = 2 N .J ′′ = 22 − m Z K (1+ σ ) ρ ∩ [ u> ¯ S σ ] u ( x, 0) ¯ S − m σ − u ( x, − m m dx ≤ − m ¯ S − m σ − M − m m S σ = 22 − m M m − ¯ S m σ mM m ρ N m S − m σ . As in the interval (0 , M ] the function f ( s ) = M m − s m mM m is convex, we can apply Jensen’s inequality and conclude that J ′′ ≤ − m ρ N m S − m σ − Z K (1+ σ ) ρ M m − u ( x, s ) m mM m dx γ Λ m , ρ N m S − m σ . Therefore, J ≤ γ (1 + Λ m , ) ρ N m S − m σ . As for J we have J = γσ ρ Z Z Q ∩ [ u> ¯ S σ ] u m ¯ S mσ u m − ¯ S m σ mu m ! dx dτ ≤ γσ ρ S mσ Z Z Q ∩ [ u> ¯ S σ ] u m M m − ¯ S m σ mM m ! dx dτ ≤ γσ ρ S mσ M m − ¯ S m σ mM m ! t sup <τ Let u be a non-negative, local, weak solution to the singularequations (4.5)–(4.6), in E T . There exists a positive constant γ depending onlyon the data { N, C o , C } , such that for all cylinders K ρ ( y ) × [ s, t ] ⊂ E T , and all σ ∈ (0 , , ρ Z ts Z K ρ ( y ) | A ( x, τ, u, Du ) | dx dτ ≤ γσ (Λ m , + Λ m , ) (cid:18) t − sρ λ (cid:19) S mσ + γ (1 + Λ m , ) (cid:18) t − sρ λ (cid:19) S m +12 σ roof - Assume ( y, s ) = (0 , Q = K ρ × (0 , t ]. By the structureconditions of A ρ Z t Z K ρ | A ( x, τ, u, Du ) | dx dτ ≤ C ρ Z Z Q u m − | Du | dx dτ ≤ C ρ (cid:18)Z Z Q u m − | Du | dx dτ (cid:19) (cid:18)Z Z Q u m dx dτ (cid:19) ≤ γ C ρ (cid:20) (1 + Λ m , ) ρ N m S − m σ + 1 σ ρ (Λ m , + Λ m , ) S m σ t ρ N (1 − m ) (cid:21) × " t sup <τ Proof of Proposition 4.1 We conclude as in the proof of Proposition 2.1, relying on Corollary B.1, insteadof Corollary 6.1. References [1] M.A. Herrero and M. Pierre, The Cauchy problem u t = ∆ u m when 0 Discrete Con-tinuous Dynamical Systems Ser. B, 17(6), 2012, 1841–1858. [3] E. DiBenedetto, U. Gianazza and N. Liao, Logarithmically SingularParabolic Equations as Limits of the Porous Medium Equation, NonlinearAnalysis Series A: Theory, Methods & Applications, 75(12), 2012, 4513–4533. [4] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s Inequality for De-generate and Singular Parabolic Equations , Springer Monographs in Math-ematics, Springer-Verlag, New York, 2012.[5] E. DiBenedetto, Y.C. 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