REGULARITY OF SOLUTIONS TO DOUBLY NONLINEAR DIFFUSION EQUATIONS

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1 Seventh Mississii State - UAB Conference on Differential Equations and Comutational Simulations, Electronic Journal of Differential Equations, Conf , ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu REGULARITY OF SOLUTIONS TO DOUBLY NONLINEAR DIFFUSION EQUATIONS JOCHEN MERKER Abstract. We rove under weak assumtions that solutions u of doubly nonlinear reaction-diffusion equations u = u m 1 + fu to initial values u0 L a are instantly regularized to functions ut L ultracontractivity. Our roof is based on a riori estimates of ut rt for a time-deendent exonent rt. These a riori estimates can be obtained in an elementary way from logarithmic Gagliardo-Nirenberg inequalities by an otimal choice of rt, and they do not only imly ultracontractivity, but rovide further information about the long-time behaviour. 1. Introduction Let us consider the quasilinear arabolic equation u = u m 1 + fu on 0, R n. Here u m 1 := u m 2 u denotes signed ower, u := div u 2 u is the -Lalacian and f is a nonlinearity deending on u. In the semilinear case = 2, m = 2, it is well-known that initial values u0 L a are instantly regularized to ut L t > 0. This roerty called ultracontractivity is imortant, because once ut L has been established, in a second ste often Hölder continuity and differentiability of ut can be shown. Surrisingly, also in the quasilinear case 2, m 2, the generated semigrou is ultracontractive. This roerty can be roved by Moser iteration see [4, 10]; i.e., ste-by-ste ut ri C i is shown for some increasing sequence of indices r i, while the constants C i are controlled so that r i and ut L can be guaranteed. However, it is much more favourable to rove ultracontractivity by a riori estimates of the time-deendent Lebesgue norm ut rt obtained from logarithmic Gagliardo-Nirenberg inequalities. A discussion of such logarithmic inequalities of 2000 Mathematics Subject Classification. 35K65, 35B35, 46E35, 35B45. Key words and hrases. -Lalacian; doubly nonlinear evolution equations; ultracontractive semigrous; logarithmic Gagliardo-Nirenberg inequalities. c 2009 Texas State University - San Marcos. Published Aril 15,

2 186 J. MERKER EJDE/CONF/17 Sobolev tye can be found in [2, 11], and in [5, 3] ultracontractivity was roved in the urely diffusive case f = 0 by this method. In [9] a rather elementary exosition of this method was given. There the method was alied to doubly nonlinear diffusion equations without a nonlinearity, i.e. to the case f = 0, and otimal results were obtained by choosing rt in an otimal way. Here it is shown how nonlinearities f can be incororated, and again comared to Moser iteration the method of time-deendent exonents shows advantages: It is indeendent of the domain and boundary conditions, it allows to handle different tyes of nonlinearities in a flexible and unified way, and the estimates of the time-deendent norm ut rt are as otimal as the regiven estimates of the nonlinearity. In [7] res. [12] similar results are obtained in secial cases, namely the 1- homogeneous case m = without nonlinearity res. the -diffusion case m = 1 with nonlinearity. However, their a riori estimates are based on logarithmic Sobolev inequalities, while for the general case of doubly nonlinear reaction-diffusion equations you need logarithmic Gagliardo-Nirenberg inequalities Outline. Let us summarize this aer: In the second section we describe how the method of time-deendent exonents can be alied to doubly nonlinear reaction-diffusion equations. We formulate logarithmic Gagliardo-Nirenberg inequalities and rove an ordinary differential inequality deending on the estimates of the nonlinearity for the time-deendent Lebesgue norm ut rt of a solution ut, where the variable exonent rt is arbitrary. Due to the validity of the differential inequality, ut rt is bounded by the solution ht of the associated differential equation to the initial value u0 a. This solution ht deends on rt, and to obtain otimal bounds of ut rt, we minimize ht w.r.t. the time-deendent exonent rt. Now, if there are minimizers rt which blow u in arbitrarily short time while ht stays bounded, then an otimal ultracontractivity estimate of ut can be roved. Because this rocedure is rather general, in the third section we discuss the case of sublinear nonlinearities in detail. Esecially, this examle shows that a recalculation of the minimizer r gives better a riori estimates than using the otimal time-deendent exonent obtained in [9] for doubly nonlinear diffusion equations without nonlinearity. The main result about doubly nonlinear diffusions with sublinear nonlinearity is Theorem 1.1. Let n 2, 1, m > 1, a > 0 and assume the validity of a > max 1, n 1 m. Let f be sublinear; i.e., there is a constant H > 0 such that fx, uu H u 2 for all x and u, then every strong solution ut L a R n of u = u m 1 +fu to an initial value u0 L a R n is instantely regularized to a function ut L R n, t > 0. More recisely, there is a constant C n,,m,a such that ut C n,,m,a u0 a a+nm a exhm T n 2 a+nm n 2 exhm T Hm T + 1 ex, a + nm exhm T and consequently the global attractor is contained in a bounded set in L.

3 EJDE-2009/CONF/17 REGULARITY OF SOLUTIONS Remarks. During the text for the reader s convenience ut L a is assumed to be a strong solution, but the method also works for weak solutions, as the arguments in the roof of [12, Lemma 4.2] show. Regarding the existence of weak solutions, let us oint out that at least in the doubly degenerated case > 2, m > 2, weak solutions with ut L m exist. For bounded domains this is roved e.g. in [1, 4, 10], a roof by Faedo-Galerkin method for general domains will be resented in a forthcoming aer. Further, it is not essential that the equation is considered on R n. In fact, the indeendence on the domain is a secial feature of the method of time-deendent exonents, so that analogous results are valid for bounded domains Ω R n with Dirichlet or Neumann boundary, and Riemannian manifolds with an Euclidean tye Sobolev inequality. However, ay attention to the fact that often the estimates of the nonlinearity f deend on the domain. 2. A riori estimates Our roof of ultracontractivity of doubly nonlinear diffusion equations relies on logarithmic Gagliardo-Nirenberg inequalities and the regiven estimates of the nonlinearity. Therefore, let us formulate logarithmic Gagliardo-Nirenberg inequalities, and let us combine these inequalities with the estimates of the nonlinearity to obtain a differential inequality for the time-deendent norm ut rt of a solution ut Logarithmic Gagliardo-Nirenberg inequalities. Logarithmic Gagliardo- Nirenberg inequalities can be used to estimate the diffusive art u m 1 of doubly nonlinear diffusion equations. Lemma 2.1 Logarithmic Gagliardo-Nirenberg inequalities. The inequalities u q u q 1 u q log q u q dx q 1 q/ log Cn,,q q u q u q q are valid for arameters 1 <, 0 < q < with q/ < 1 and functions u L q on R n with u L. Hereby the constant C deends on n and only in the case < n, and on n, and a finite uer bound of q in the case n. An elementary roof of these inequalities is given in [9]. Now choose 2 /q instead of q and substitute u with u q/ to obtain the reformulation u u u log u dx q 2 / log Cn,,q u q/ u q valid for arameters 1 <, 0 < q < with q > 2 / and functions u L with u q/ L. Equivalently, these inequalities can be formulated in the arametric form u u u log u dx µ uq/ u q 2 / log C n,,q eq 2 / µ valid for all µ > 0 under the same conditions as before. + log u q 2.1

4 188 J. MERKER EJDE/CONF/ A differential inequality for the time-deendent norm. The basic differential inequality for the time-deendent Lebesgue norm ut rt is obtained from d dt u r = u r ṙ r 2 log = u r ṙ r 2 u r 1 + r u r ṙ log u + r u u u r u 2 u r u r 2.2 u r log u r + r2 ṙ u r uu r 1 and u = u m 1 +fu by alying logarithmic Gagliardo-Nirenberg inequalities to estimate the diffusive art u m 1, and the regiven estimates of the nonlinearity f to estimate the reactive art. To estimate the diffusive art, note that the -Lalacian satisfies u m 1 u u r 1 = m 1 2 u m 1 u r 1 = m 1 r = m 1 r r + m u r+m u u r+m 1 1 1/. Thus by substituting w := u r/ and with the abbreviation q := r + m 1 /r the bracket in equation 2.2 is u to the reactive art exactly the left hand side w w w log w m 1 r 2 r w q/ ṙ r + m w of the arametric form of logarithmic Gagliardo-Nirenberg inequalities 2.1 with arameter µ given by µ = m 1 r 2 r ṙ r + m. Hence we imose the restriction q > 2 / and aly the logarithmic Gagliardo- Nirenberg inequalities in their arametric form along with w q to conclude d dt u ṙ r u r r 2 q 2 / fuu r 1 +. u r 1 r log C eq 2 / µ = u r q/ r r q + log u r This differential inequality deends strongly on the estimate of the nonlinearity f: If the nonlinearity satisfies an estimate of the form fuu r 1 Hr, u r u r r with a function H, then the differential equation ḣ = F th logh + Gth + Hr, hh 2.3 corresonds to the differential inequality, and its solution ht to the initial value u0 a is a bound for ut rt. Hereby the functions and Gt := ṙ r 2 F t := ṙ r q q 2 / = q 2 / log C eq 2 / µ n1 m ṙ rr + nm

5 EJDE-2009/CONF/17 REGULARITY OF SOLUTIONS 189 nṙ = rr + nm C ṙr + m log e m 1 rr r1 / + m are the same as in [9]. However, even if fuu r 1 can not be estimated by u r only, but in terms of u r and u rq/2 via an inequality of the form fuu r 1 ɛ u rq/2 + gr, ɛ, u r u r r, then as long as ɛ is so small that µ r2 ṙ ɛ is ositive, the diffusive art comensates the reactive art estimated by ɛ u rq/2. Therefore, it is ossible to derive a more comlicated differential inequality for ut rt involving not only rt but also ɛt as time-deendent arameter see [12] for an examle where this situation is handeled successfully. For simlicity, in the following let us mainly discuss the case where the differential equation corresonding to the inequality has the form 2.3, although the general rocedure does not deend strongly on the form of this ODE. As ut rt is bounded by the solution ht to the initial value u0 a, let us minimize h w.r.t. r and ossibly ɛ to obtain an otimal bound of u r. It deends on the comlexity of the ODE for h how this is done in detail: If the ODE for h can be solved exlicitly, then ht deends on r and ṙ via an integral T Lr, ṙ dt. Thus minimizing ht is equivalent to solving the Euler-Lagrange 0 equations associated to this integral. Unfortunately, the associated differential equation 2.3 often can not be solved analytically for general nonlinearities f, so there is no formula exressing ht in terms of h0, r and ṙ. But still it is ossible to minimize h on the fly by an otimal choice of r. In fact, denote the differential equation 2.3 by ḣ = F rt, ṙt, ht, then ht = h0+ T F rt, ṙt, ht dt and minimizing ht is equivalent to the 0 minimization of T F rt, ṙt, ht dt w.r.t. r under the constraint that h solves 0 the differential equation ḣ = F r, ṙ, h. Because of dh = F rt, ṙt, ht dt, the infinitesimal deendence of h on r is given by dh dr = F r,ṙ,h ṙ. Thus the corresonding Euler-Lagrange equations for r are d dt F ṙ = F r +F h F/ṙ. Hence the comlete system of ODEs for the variable h and the control arameter r is ḣ = F r, ṙ, h d dt F ṙ = F r + F h F, ṙ and you are interested in solutions ht, rt to the initial value h0 = u0 a and the boundary values r0 = a and rt = b. In fact, then ut b ht, where the right hand side deends on u0 a, a, b and T only. This is the a riori estimate we searched for, because if the right hand side stays bounded as b, an inequality of the form ut C u0 a, a, T has been roved. This inequality imlies that in the arbitrary small time T the function u0 L a is regularized to a function ut L, i.e. ultracontractivity has been shown. Moreover, the deendence of the right hand side on T gives you information about the long-time behaviour. For examle, if the right hand side converges to 0 for

6 190 J. MERKER EJDE/CONF/17 T, every solutions u aroaches 0, and if the right hand side is bounded in T, you have a global attractor in L. Although it seems unlikely to find an exlicit solution h and minimizer r for general estimates of nonlinearities, in [9] an exlicit solution is calculated for the urely diffusive case, and in the next section we succeed to calculate an exlicit solution for sublinear nonlinearities. But even if you do not find an exlicit solution, then still you can try to estimate the right hand side of 2.3 by a right hand side for which you can solve the ODE, and although the obtained estimate for u is worse, it may be enough to conclude ultracontractivity. A further alternative is to study the ODE numerically. Let us end here our general descrition of the method of time-deendent exonents alied to doubly nonlinear diffusions. As a worthwhile examle in the next section the articular case of a sublinear nonlinearity is discussed in detail. 3. Sublinear Nonlinearities Sublinear nonlinearities rovide an examle for the fact that recalculating the minimizer r gives better estimates than simly using the minimizer rt = 1 n1 m 1 1 At+B + obtained in [9] for doubly nonlinear diffusion equations without a nonlinearity. A nonlinearity f is called sublinear, if there is a constant H > 0 such that fx, uxux r 1 dx H u r r holds for all r > 0 and u L r. Particularly, a Caratheodory function with fx, uu H u 2 for all x and u is sublinear. In this case Hr, h H is constant, and equation 2.3 is equivalent to logh = F t logh + Gt + H. Under the boundary conditions r0 = a, rt = b, this equation has the exlicit solution ht = h0 ab+nm b + nm ba+nm ex bn T 0 n 2 ṙ r + nm 2 nc r + m ṙ log e m 1 rr r + nm nr + H r + nm dt 3.1 Now for the otimal time-deendent exonent rt = 1 At+B + n1 m 1 1 of the doubly nonlinear diffusion equation without a nonlinearity the last term in the integral is nr H r + nm = H n H n2 m At + B 2,

7 EJDE-2009/CONF/17 REGULARITY OF SOLUTIONS 191 and thus an additional factor ex H n T H n2 m 1 1 1AT 2 /2+BT the calculation of ht in [9]. With B = a + nm AT = this factor becomes n ex a b a + nm b + nm. nm a + b + 2nm 2a + nm b + nm and converges as b to n nm ex 1 HT. 2a + nm arises in HT Thus with the otimal time-deendent exonent for ure diffusions hyercontractivity and ultracontractivity of solutions can be roved also in the resence of sublinear reaction terms, but merely by an exonential estimate in time ut C n,,m,a u0 n ex 1 a a+nm a T n a+nm nm 2a + nm HT. Esecially, this estimate does not rovide any useful information about the longtime behaviour of the reaction-diffusion equation. If instead, we recalculate the otimal time-deendent exonent, the Euler-Lagrange equation corresonding to the minimization of the integral in the exlicit solution ht is very similar to the equation obtained in [9]. In fact, by similar simlifications as there the Euler-Lagrange equations are equivalent to r ṙ = 2 ṙ + Hm r + nm and thus to d dt logṙ = 2 d logr + nm + Hm. dt Hence we obtain nm ṙ = A exhm tr + 2 with a constant A. Integrating this equation gives 1 r + nm A = Hm with a constant B, so that finally rt = exhm t B Hm Hm nm A exhm t + B is obtained. The boundary conditions r0 = a and rt = b imly A exhm T

8 192 J. MERKER EJDE/CONF/17 and = H 2 m a b a + nm b + nm B = Hm b + nm + b a exhm 1 1 1T 1 a + nm b + nm Thus minimizers blow u e.g. if m > 1, as then A < 0, B > 0. Now let us calculate ht for the recalculated minimizers r. The integral in formula 3.1 contains as first term n 2 ṙ r + nm 2 = n2 A exhm t, 2 the second term is log = log A nc r+m ṙ e m 1 1 rr 1r+nm nc Hm e 2 m 1 1 A exhm 1 1 1t+B 1 exhm 1 1 1t Hm nm 1 1 1A exhm 1 1 1t+B Hm nm 1 1 1A exhm 1 1 1t+B Hm nm A exhm 1 1 1t+B and the third term is nr H r + nm = n H na exhm t + B. Thus the integral in 3.1 is T n2 exhm t A 2 0 log A + n nc Hm e 2 m 1 1 A exhm 1 1 1t+B 1 exhm 1 1 1t Hm nm 1 1 1A exhm 1 1 1t+B Hm nm 1 1 1A exhm 1 1 1t+B Hm nm A exhm 1 1 1t+B H na exhm t + B dt Substitute s = exhm t, then the differential element is given by ds = Hm exhm t dt, and we obtain n2 2 A exhm 1 1 1T Hm log A nc Hm 1 1 1s e 2 m 1 1 As+B 1 Hm nm 1 1 1As+B Hm nm 1 1 1As+B Hm nm As+B = n2 2 A Hm exhm 1 1 1T 1 log + logs + log Hm + nm As + B logas + B log Hm + n H nas + B ds A nc Hm e 2 m 1 1

9 EJDE-2009/CONF/17 REGULARITY OF SOLUTIONS 193 nm As + B log Hm nm + As + B + n H nas + B ds Now comute the integrals of each term and use Hm A + B = a + nm Hm A exhm T + B = b + nm to obtain for the integral in 3.1 the exression n 2 b a a + nm b + nm log log e Hm 1 1 1T H 2 nc m b a e 2 3 m 1 1 a+nm 1 1 1b+nm n 2 b a a+nm 1 1 1b+nm exhm 1 1 1T Hm 1 1 1T 1+1 exhm 1 1 1T nm b+nm log Hm 1 + nm b+nm nm a+nm log Hm 1 + nm a+nm n 2 nm n2 2 1 b a H 2 m a+nm 1 1 1b+nm n 1 m nm b+nm log Hm 1 nm b+nm nm a+nm log Hm 1 nm a+nm n 1 m nm b+nm log Hm 1 nm b+nm nm a+nm log Hm 1 nm a+nm Hm 1 1 1b a m a+nm 1 1 1b+nm n 3 1 n 4 Hm b+a+nm b a a+nm b+nm

10 194 J. MERKER EJDE/CONF/17 Only the first two terms deend on T, and because all the other terms converge for b to a constant deending on n,, m and a, we obtain the estimate ut C n,,m,a u0 a a+nm a exhm T n 2 a+nm ex n 2 exhm T Hm T + 1 a + nm exhm T This estimate roves our main theorem. Note that for large T the last factor increases like n 2 m ex H a + nm T, while the first factor decreases like n 2 m ex H a + nm T in T. Secially, for every ɛ > 0 there is a constant D not deending on time T but on the indices n,, m and a and the norm of the initial value u0 a such that ut D for all T ɛ. Therefore, after an arbitrarily short time the function ut is contained in a bounded set in L. This estimate is much better than the one obtained before by using the otimal time-deendent exonent of the urely diffusive case. In articular, it shows that the global attractor of doubly nonlinear diffusions with sublinear reaction terms lies in L. Acknowledgements. I wish to thank Peter Takáč for his valuable comments and suggestions. This research was suorted by the Deutsche Forschungsgemeinschaft DFG, Germany. References [1] Hans Wilhelm Alt, Stehan Luckhaus, Quasilinear Ellitic-Parabolic Differential Equations, Mathematische Zeitschrift , [2] D. Bakry, T. Coulhon, M. Ledoux, L. Saloff-Coste, Sobolev Inequalities in Disguise, Indiana University Mathematics Journal 44, No , [3] M. Bonforte, G. Grillo, Suer and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Math. Iberoamericana 22, no , [4] Chen Caisheng, Global Existence and L Estimates of Solution for Doubly Nonlinear Parabolic Equation, Journal of Mathematical Analysis and Alications , [5] F. Ciriani, G. Grillo, Suer and ultracontractive bounds for doubly nonlinear evolution equations, Journal of Differential Equations , [6] Manuel Del Pino, Jean Dolbeaut, The otimal Euclidean L -Sobolev logarithmic inequality, Journal of Functional Analysis , [7] Manuel Del Pino, Jean Dolbeaut and Ivan Gentil, Nonlinear diffusions, hyercontractivity and the otimal L -Euclidean logarithmic Sobolev inequality, Journal of Mathematical Analysis and Alications , [8] Ivan Gentil, The general otimal L -Euclidean logarithmic Sobolev inequality by Hamilton- Jacobi equations, Journal of Functional Analysis , [9] Jochen Merker, Generalizations of logarithmic Sobolev inequalities, Discrete and Continuous Dynamical Systems - Series S, Volume 1, Number , [10] M. M. Porzio, L loc estimates for a class of doubly nonlinear arabolic equations with sources, Rendiconti di Matematica, Serie VII, , [11] Laurent Saloff-Coste, Asects of Sobolev-Tye Inequalities, London Mathematical Society Lecture Note Series, Cambridge University Press,

11 EJDE-2009/CONF/17 REGULARITY OF SOLUTIONS 195 [12] Peter Takac, L -Bounds for Weak Solutions of an Evolutionary Equation with the - Lalacian, Proceedings of the international conference Function Saces, Differential Oerators and Nonlinear Analysis, FSDONA 2004, B-Print 2005, Jochen Merker University of Rostock, D Rostock, Germany address: jochen.merker@uni-rostock.de