University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2012 Decay Estimates for Nonlinear Wave Equations with Variable Coefficients Michael Jacob Roberts mrober33@utk.edu Recommended Citation Roberts, Michael Jacob, "Decay Estimates for Nonlinear Wave Equations with Variable Coefficients. " Master's Thesis, University of Tennessee, 2012. http://trace.tennessee.edu/utk_gradthes/1330 This Thesis is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.edu.
To the Graduate Council: I am submitting herewith a thesis written by Michael Jacob Roberts entitled "Decay Estimates for Nonlinear Wave Equations with Variable Coefficients." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. We have read this thesis and recommend its acceptance: Henry Simpson, Don B. Hinton (Original signatures are on file with official student records.) Grozdena H. Todorova, Major Professor Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School
Decay Estimates for Nonlinear Wave Equations with Variable Coefficients A Thesis Presented for The Master of Science Degree The University of Tennessee, Knoxville Michael Jacob Roberts August 2012
c by Michael Jacob Roberts, 2012 All Rights Reserved. ii
Acknowledgements In a large way, this paper would have not been possible without the aid of Dr. Grozdena Todorova, my thesis adviser. She has been invaluable throughout the entire process, sticking with me even when I m at my worst. Further, Dr. Borislav Yordanov was a great help, taking time to aid me when needed. iii
Abstract We studied the long time behavior of solutions of nonlinear wave equations with variable coefficients and an absorption nonlinearity. Such an equation appears in models for traveling waves in a non-homogeneous gas with damping that changes with position. We established decay estimates of the energy of solutions. We found three different regimes of decay of solutions depending on the exponent of the absorption term. We show the existence of two critical exponents. For the exponents above the larger critical exponent, the decay of solutions of the nonlinear equation coincides with that of the corresponding linear problem. For exponents below the larger critical exponent, the solution decays much faster. Lastly, the subcritical region is further divided into two subregions with different decay rates. Deriving the sharp decay of solutions even for the linear problem with potential is a delicate task and requires serious strengthening of the multiplier method. Here we used a modification of an approach of Todorova and Yordanov to derive the exact decay of the nonlinear equation. iv
Contents 1 Introduction 1 2 The Weighted Energy 7 3 Definitions of φ and θ 17 4 Large p Case 20 5 Small p Case 22 6 Combined Results 27 Bibliography 29 Vita 32 v
Chapter 1 Introduction In this thesis, we consider the following dissipative non-linear wave equation u tt div(b(x) u) + a(x)u t + u p 1 u = 0, x R n, t > 0 (1.0.1) u(0, x) = u 0 (x) H 1 (R n ), u t (0, x) = u 1 (x) L 2 (R n ) where n 3, 1 < p < (n + 2)/(n 2), a C 0 (R n ), and b C 1 (R n ) are positive, and u 0 and u 1 have compact support such that u 0 (x) = 0 and u 1 (x) = 0 for x > R. In addition, we require that a(x) and b(x) behave in such a way that b 0 (1 + x ) β b(x) b 1 (1 + x ) β (1.0.2) and a 0 (1 + x ) α a(x) a 1 (1 + x ) α (1.0.3) where α, β R and a 0, a 1, b 0, b 1 are positive constants and α < 1, 0 β < 2, 2α + β < 2. (1.0.4) 1
This problem models a wave traveling with displacement u(x, t) in a nonhomogenous medium with space-dependent friction coefficient a(x) and bulk modulus b(x), which accounts for varying temperature in the medium, along with a nonlinearity, causing the system to be much more complicated than previous efforts that only focused on more homogeneous cases. Until now, the significant issues in this equation have only been investigated separately. Combining the variable coefficients a(x) and b(x) with the nonlinearity makes this problem new and interesting. The restrictions on the exponents α < 1, 0 β < 2, 2α + β < 2 are natural because as β 2 and α 0+, the decay approaches infinity (see Theorem 1.0.3). Further, if α 1, there is no longer decay, and the energy instead dissipates to some positive constant per Mochizuki [1976/1977]. It is possible that instead of being defocusing, the nonlinearity could be focusing, with a negative coefficient. In the paper of Ikehata, Todorova, and Yordanov [2009], they show that for supercritical nonlinear exponent p > 1 + 2, small initial data n α solutions are global, but for subcritical p 1 + 2, the solutions blow up in finite n α time for data positive in average. Luckily, with a defocusing nonlinearity, the global existence of the solution is a classical result. The asymptotic behavior of the energy is still under investigation, however. cases. Past works similar to (1.0.1) are easily separated into the linear and nonlinear Khader [2011], Nishihara [2010], Lin, Nishihara, and Zhai [2010], Lin, Nishihara, and Zhai [2011], and Todorova and Yordanov [2007] each studied the nonlinear case with an absorption nonlinearity. None of these papers attempt to estimate asymptotic decay rates for the case with the variable coefficient b(x), however. They each assume b(x) = 1, which simplifies the calculations and results. Kenigson and Kenigson [2011] consider a linear case similar to (1.0.1) but with a space-time dependent damping coefficient and no b(x), and in the paper of Radu, 2
Todorova, and Yordanov [2009], the Laplacian is split by the b(x), but with a critical difference in that their equation is linear as well. The nonlinearity raises many issues that must be managed very delicately, so including it is a worthy expansion of the problem. Mathematically, the primary difficulties arise in dealing with the energy terms that come from the interactions among the nonlinearity, a(x), and b(x). Using an advanced weighted multiplier method developed by Todorova and Yordanov, we overcome these issues. First, we must create what we hope will be an approximate solution to (1.0.1), using the following conjecture. Conjecture 1.0.1. Under the assumptions (1.0.2) and (1.0.3), there exists a subsolution A(x) which satisfies a related differential inequality div(b(x) A(x)) a(x) (1.0.5) with the following properties, for d 0, d 1 > 0 : d 0 (1 + x ) 2 α β A(x) d 1 (1 + x ) 2 α β (1.0.6) a(x)a(x) µ = lim inf > 0. x b(x) A(x) 2 (1.0.7) There are multiple cases with only mild restrictions on a(x) and b(x) that grant existence of such subsolutions. These cases and that µ = n α 2 α β are dealt with in Radu, Todorova, and Yordanov [2009]. For now, we assume A(x) exists, which we then use to construct, with δ (0, µ/2), σ(x) = (µ δ)a(x). (1.0.8) Let us mention that σ(x) ultimately plays a crucial role in the definitions of the multiplier weights. The idea is to imitate similar methods used to approximate 3
the solutions of linear equations with constant coefficients with the diffusion phenomenom, through which, it is shown that the solution of the linear dissipative equation u tt u + u t = 0 has similar large time behavior to the solution of the diffusion equation w t w = 0. In the linear case with constant coefficients, (1.0.5) becomes the Poisson equation A(x) = 1 with radial, nonnegative solution A(x) = x 2. This A(x) is then used 2n to construct the Gaussian approximate solution w(x, t) = t n 2 e x 2 4t. In our case, with variable coefficients and a nonlinearity, the diffusion phenomenon has not been proven, but we still use σ(x) to construct our approximate solution φ(x, t) = t m e σ(x) t, with suitable parameter m to optimize decay, in the hopes that φ and u, the solution of (1.0.1), will have similar large t behavior. Before we begin, it is important to note that supp(u(t, x)) is contained in the set {x R n : x [(1 + R) (2 β)/2 + t b 1 ] 2/(2 β) }. Notice that the speed of propogation is finite, but variable due to the coefficient b(x). This and the following proposition are proven in Radu, Todorova, and Yordanov [2009]. Proposition 1.0.2. Define g(t) := inf{a(x) : x supp u(, t)}, (1.0.9) G(t) := sup{a(x) : x supp u(, t)}, (1.0.10) 4
and 2α, if β 2 2α, 2 β γ = 0, if α 0. (1.0.11) Then g(t) g 0 t γ if t t 0, (1.0.12) where g 0 and G 0 are positive constants. G(t) G 0 t 2 γ if t t 0, (1.0.13) At first, we factor out our approximate solution from our solution u in the hopes that the new equation will have more timid large time behavior. We then apply a strengthened multiplier method using weights designed for our problem, progressing through the proofs by simply placing sufficient conditions on the arbitrary weights in order to grant us important mathematical qualities of the solution of the new equation. Estimating the energy decay of this altered problem gives us a strict energy decay estimate for the original problem, and lastly, we verify that weights with such qualities actually exist. After doing all this, we obtain the following theorem concerning the decay rates of the energy, E(t) = 1 2 u 2 t + b u 2 dx + 1 p + 1 u p+1 dx. Theorem 1.0.3. The energy of the solution of (1.0.1) satisfies, for some c > 0 and all t 0, E(t) ct m 1 5
where, for δ > 0, 2 p 1 δ m = 2 + α p+1 p 1 n p 1 2 α β 2 β if 1 < p 1 +, n α 2 β δ if 1 + < p n+α, n α n α n α δ if n+α < p < n+2. 2 α β n α n 2 Remark 1.0.4. Notice that as the nonlinear exponent p becomes larger, the nonlinearity affects the decay less. For large p, the optimal decay corresponds with the decay of the linear equation, derived in Radu, Todorova, and Yordanov [2009]. This paper is organized such that in chapter 2, we derive the main inequalities for the weighted energy followed by the corresponding inequalities for the unweighted energy. In chapter 3, we define the weights φ and θ and derive the main energy decay inequality. In chapter 4, we prove a theorem concerning the rate of decay for large p, following which, in chapter 5, we prove a similar theorem for small p. Lastly, in chapter 6, we prove the main theorem 1.0.3. 6
Chapter 2 The Weighted Energy To begin, we factor out the approximate solution φ = t m e σ(x) t, granted by imitating the diffusion phenomenon of linear equations, from u. Setting u = vφ gives a new partial differential equation with respect to v. Ideally, this new equation will have simpler large time behavior. We get v tt ˆb 1 v ˆb 2 v + â 1 v t + â 2 v + φ p 1 v p 1 = 0 (2.0.1) with new coefficients ˆb1 (x) = b(x), ˆb2 (x, t) = b(x) + 2b(x)φ(x, t) 1 φ(x, t) (2.0.2) â 1 (x, t) = a(x) + 2φ(x, t) 1 φ t (x, t) â 2 (x, t) = φ(x, t) 1 (φ tt (x, t) div(b(x) φ(x, t)) + a(x)φ t (x, t)). Now we apply a strengthened multiplier method using weights that will later be specifically designed for our problem. Note that despite one weight s being named φ, we do not mean to necessarily imply a connection between the weight and the approximate solution right now. This naming is used only in foresight that the two are actually the same. 7
Proposition 2.0.5. Using multipliers φv and θv t and adding the resulting equations together gives the weighted energy identity d dt E w + F + G + H = 0, where E w = 1 θ(vt 2 + b v 2 ) + 2φv t v + (â 2 θ + φ t + aφ)v 2 + φp 1 θ v p+1 dx, 2 p + 1 F = 1 ( θ t + 2(a + 2φ 1 φ t )θ 2φ)vt 2 dx + b( θ 2θφ 1 φ) v t vdx 2 + 1 b( θ t + 2φ) v 2 dx, 2 G = 1 [â 2 φ (â 2 θ) t ]v 2 dx, 2 H = [φ p 1 1 + p (θφp 1 ) t ] v p+1 dx. Proof. Using the finite speed of propagation and elementary calculus allows us to integrate by parts over the compact support. result. Doing so straightforwardly yields the desired We now seek to bound the weighted energy E w so that the unweighted energy will be decaying. In order to proceed, we place conditions on our weights θ and φ. The proof that there are weights satisfying these conditions will come later. Proposition 2.0.6. Assume that θ > 0 and φ > 0 are C 1 -functions such that (i) θ t + φ 0 (ii) ( θ t + 2θ(a + 2φ 1 φ t ) 2φ)( θ t + 2φ) b θ 2θφ 1 φ 2. Then F 0 for t 0. Proof. This follows directly from the quadratic form of F in v t and v. 8
Proposition 2.0.7. Assume that θ and φ satisfy conditions (i) and (ii) in Proposition 2.0.6, and in addition satisfy the following two conditions: (iii) (iv) (p + 1)φ p (θφ p 1 ) t φ p â 2 φ (â 2 θ) t k 0 C φ with C (x, t) that φ 1 C p+1 p 1 dxdt < and k 0 > 0. t 0 Then, for t 0 and some k 0, we have G + H k φ 1 C p+1 p 1 dx. Proof. Using (iii), (iv), and Holder s inequality with exponents p+1 p+1 and yields p 1 2 G + H = 1 [â 2 φ (â 2 θ) t ]v 2 dx + 2 k 0 C φv 2 dx + 2 p 1 k 0 φ 1 C p+1 p 1 dx. 2(p + 1) [φ p 1 p + 1 (θφp 1 ) t ] v p+1 dx 1 p + 1 φp v p+1 dx Letting k = k 0 (p 1)/(2p + 2) completes the proof. These first four conditions, when applied to the weighted energy identity d E dt w + F + G + H = 0, give us a constant upper bound on the weighted energy, which is one step in ensuring the decay of the unweighted energy, as follows: Theorem 2.0.8. If conditions (i)-(iv) hold, then E w (t) E w (t 0 ) + k t 0 φ 1 C p+1 p 1 dxdt <. 9
that Proof. Using Propositions 2.0.6 and 2.0.7 in the weighted energy equality gives d dt E w = F (G + H) (G + H) k φ 1 C p+1 p 1 dx. Finally, integrating both sides from t 0 to and using the integral inequality in (iv) leaves us with a finite bound on the weighted energy as claimed. We now need to eliminate the terms that lack an obvious sign by imposing three more conditions, and also, for our next theorem, we will need the following lemma from Radu, Todorova, and Yordanov [2009]: Lemma 2.0.9. If f C([t 0, )) is a positive function, then e t t t f(s)ds s 0 t e f(τ)dτ 1 0 ds max t 0 s [t 0,t] f(s). Theorem 2.0.10. Given that conditions (i)-(iv) hold and that (v) C â 2 satisfying sup t t 0 θφ 2 C p+1 p 1 dx < holds, then for t t 0 and for some k 0, k 1 > 0, we have that φv 2 dx k 0 + k 1 t 2α/(2 β). Proof. Recall the weighted energy: E w = 1 2 θ(v 2 t + b v 2 ) + 2φv t v + (â 2 θ + φ t + aφ)v 2 + 1 p + 1 φp 1 θ v p+1 dx. 10
As shown in Theorem 2.0.8, E w (t) b 3 := E w (t 0 ) + k t 0 φ 1 C p+1 p 1 dxdt <. After rearranging terms in E w, we have that d dt φv 2 dx + a(x)φv 2 dx 2b 3 + â 2 θv 2 dx 2b 3 + 2c 1 θφ 2 C p+1 p 1 dx 2b 3 + c 2 = c 0 by condition (v). 2 p + 1 φp 1 θ v p+1 dx per Young s inequality Using the finite speed of propagation and (1.0.12), we find a lower bound on a(x) for x supp(u): g 0 t 2α/(2 β) a(x). This gives the ordinary differential inequality d dt φv 2 dx + g 0 t 2α/(2 β) φv 2 dx c 0, which can be solved to show φv 2 dx e t t ] t g 0 s γ ds s 0 t g [c 1 + c 0 e 0 τ γdτ 0 ds t 0 for t > t 0 and γ as defined in (1.0.11). Because of Lemma 2.0.9 and the fact that t γ is decreasing, φv 2 dx e t t ] t g 0 s γ ds s 0 t g [c 1 + c 0 e 0 τ γdτ 0 ds t 0 c 0 e t t g 0 s γds 0 + max s [t 0,t] csγ k 0 + k 1 t γ. Thus, by the definition of γ, φv 2 dx k 0 + k 1 t 2α/(2 β) for some positive constants k 0 and k 1, as claimed. Using the previous theorem, we can eliminate some unsigned terms. 11
Lemma 2.0.11. Given conditions (i)-(v), (θ φɛt γ )(vt 2 + b v 2 ) + aφv 2 + θφp 1 v p+1 dx b 4 + p + 1 (φɛ 1 t γ φ t )v 2 dx where ɛ = (2k 1 ) 1 and b 4 is some positive constant. Proof. First, consider the unsigned term 2φv t v in the weighted energy: 2φv t v = 2φ ɛ 1/2 ɛ 1/2 t γ/2 t γ/2 v t v φɛt γ v 2 t + φɛ 1 t γ v 2 by Young s inequality φɛt γ v 2 t + φɛ 1 t γ v 2 + φɛt γ b(x) v 2. Using this inequality, we then estimate 2φv t v from below: 2φv t v 2φv t v (φɛt γ )(v 2 t + b v 2 ) φɛ 1 t γ v 2. Further, by Theorem 2.0.8 and the previous, b 3 1 θ(vt 2 + b v 2 ) + 2φv t v + (â 2 θ + φ t + aφ)v 2 + 1 2 p + 1 φp 1 θ v p+1 dx (θ φɛt γ )(vt 2 + b v 2 ) + (φ t φ ɛt γ )v2 + aφv 2 C θv 2 + 2θφp 1 v p+1 dx. p + 1 Rearranging the terms gives (θ φɛt γ )(vt 2 + b v 2 ) + aφv 2 dx b 3 + ( φ ɛt φ t)v 2 + C θv 2 2θφp 1 v p+1 dx. γ p + 1 Note that, by Young s inequality and (v), for some positive c 3, c 4, C θv 2 dx 1 p + 1 θφ p 1 v p+1 +c 3 θφ 2 C p+1 p 1 dx = 1 p + 1 θφp 1 v p+1 dx+c 4. 12
Therefore, (θ φɛt γ )(vt 2 + b v 2 ) + aφv 2 + 1 p + 1 θφp 1 v p+1 dx b 4 + (φɛ 1 t γ φ t )v 2 dx, as claimed. The previous lemma has removed most of the unsigned terms from E w. We now need to simplify the factor (θ φɛt γ ) and bound the resulting integral by a constant. The two following conditions guarantee these results. Theorem 2.0.12. Given (i)-(v) and (vi) (vii) φ k 1 t γ θ φ t k 1 t γ φ for some positive constant k 1 and sufficiently large t, it can be shown that Proof. Using (vi), 1 2 θ(v2 t + b v 2 )dx C, (2.0.3) aφv 2 dx C, (2.0.4) 1 p + 1 θφp 1 v p+1 dx C for some C > 0. (2.0.5) φ k 1 t γ θ 1 2k 1 t γ φ 1 2 θ θ ɛtγ φ 1 2 θ, letting ɛ = (2k 1) 1. Thus, by Lemma 2.0.11, 1 2 θ(v2 t + b v 2 ) + aφv 2 + 1 p + 1 θφp 1 v p+1 dx b 4 + (φɛ 1 t γ φ t )v 2 dx. 13
Further, using (vii), we have that ɛ 1 t γ φ φ t ɛ 1 t γ φ + k 1 t γ φ = 2k 1 t γ φ + k 1 t γ φ = 3k 1 t γ φ, for sufficiently large t. The previous and Theorem 2.0.10 give, for t t 0, 1 2 θ(v2 t + b v 2 ) + aφv 2 + 1 p + 1 θφp 1 v p+1 dx b 4 + 3k 1 t γ φv 2 dx b 4 + 3k 1 t γ φv 2 dx C. Note that now each term under the integral on the left hand side is positive, so we have accomplished our goal of removing the unsigned terms. We can therefore use the upper bound C for each term individually, which is what was to be shown. We now reintroduce the actual solution u by substituting v = uφ 1 back into the estimates obtained above. One last condition is needed to ensure that doing so preserves a constant upper bound. Theorem 2.0.13. Given conditions (i)-(vii) and (viii) θφ 3 (φ 2 t + b φ 2 ) k 2 a(x) for some k 2 > 0, 14
we have that aφ 1 u 2 dx K, θφ 2 u p+1 dx K, θφ 2 (u 2 t + b u 2 )dx K for some positive K. Proof. Using v = uφ 1 in (2.0.4) and (2.0.5) immediately gives two simple results: aφ 1 u 2 dx K, (2.0.6) θφ 2 u p+1 dx K. (2.0.7) Applying v = uφ 1 to (2.0.3) is a bit more complicated. Notice that v = uφ 1, v t = u t φ 1 uφ 2 φ t, vt 2 = u 2 t φ 2 2uu t φ 3 φ t + u 2 φ 4 φ 2 t, ( ) ( ) 1 1 vt 2 = 2 u2 t φ 2 3u 2 φ 4 φ 2 t + 2 u2 t φ 2 + 4u 2 φ 4 φ 2 t 2uu t φ 3 φ t. (2.0.8) Working toward producing the unweighted energy, we estimate v 2 t and v 2 from below, starting with 2uu t φ 3 φ t = (2uφ 1 φ t )u t φ 2 ( 1 2 u2 t + 1 ) 2 4u2 φ 2 φ 2 t φ 2 1 2 u2 t φ 2 + 4u 2 φ 4 φ 2 t. 15
Using this in (2.0.8), we have that v 2 t 1 2 u2 t φ 2 3φ 4 φ 2 t u 2. (2.0.9) A similar method can be used to show v 2 1 2 u 2 φ 2 3φ 4 φ 2 u 2. (2.0.10) Finally, using (2.0.9) and (2.0.10) in (2.0.3) yields 2C θ(vt 2 + b v 2 )dx 1 2 u2 t φ 2 θ 3φ 4 φ 2 t u 2 θ + 1 2 b u 2 φ 2 θ 3bφ 4 φ 2 θu 2 dx. Rearranging gives 1 2 θφ 2 (u 2 t + b u 2 )dx 2C + 3 θφ 4 u 2 (φ 2 t + b φ 2 )dx. By (viii), θφ 4 (φ 2 t + b φ 2 )u 2 k 2 a(x)φ 1 u 2. Using this, we get θφ 2 (u 2 t + b u 2 )dx 4C + 6 4C + 6 θφ 4 u 2 (φ 2 t + b φ 2 )dx k 2 aφ 1 u 2 dx K per (2.0.6). This completes the proof. 16
Chapter 3 Definitions of φ and θ Recall the conditions sufficient for energy decay for sufficiently large t: (i) θ t + φ 0 (ii) ( θ t + 2θ(a + 2φ 1 φ t ) 2φ)( θ t + 2φ) b θ 2θφ 1 φ 2 (iii) (p + 1)φ p (θφ p 1 ) t φ p (iv) â 2 φ (â 2 θ) t k 0 C φ where k 0 > 0 and t 0 (v) C â 2 satisfying sup θφ 2 C p+1 p 1 dx < t t 0 (vi) φ k 1 t γ θ φ 1 C p+1 p 1 dxdt < (vii) φ t k 1 t γ φ for some k 1 > 0 (viii) θφ 3 (φ 2 t + b φ 2 ) k 2 a(x) for some k 2 > 0 We propose the following definitions of the weights φ and θ, and then ensure the sufficient conditions are met on the support of the solution: φ(x, t) = t m e σ(x) t (3.0.1) θ(x, t) = 3 ( 6 4 t + σ(x) ) 1 φ(x, t), t 2 17
where σ(x) is defined in (1.0.8). The constants 3 and 6 are chosen for technical 4 reasons, ensuring that the eight conditions are satisfied. With these choices of weights, we have a crucial theorem. Theorem 3.0.14. Given conditions (i)-(viii) hold and that φ and θ are defined as in (3.0.1), E(t) := 1 2 u 2 t + b u 2 dx + 1 p + 1 u p+1 dx ct m 1 for some c > 0. Proof. First we look at θφ 2 : θφ 2 = 3 4 Using this and Theorem 2.0.13, we obtain ( 6 t + σ(x) ) 1 φ 1. t 2 3 θφ 2 (u 2 t + b u 2 + u p+1 )dx = 4 3 = 4 K 0. ( 6 t + σ(x) ) 1 φ 1 (u 2 t 2 t + b u 2 + u p+1 )dx ( 6 t + σ(x) ) 1 t m e σ(x) t 2 t (u 2 t + b u 2 + u p+1 )dx Rearranging yields ( 6 + σ(x) ) 1 e σ(x) t (u 2 t + b u 2 + u p+1 )dx K 1 t m 1. t Furthermore, based on a Taylor s series approximation, e σ(x) t ( ɛ 6 + σ(x) ) t for ɛ sufficiently small. This leaves 1 2 (u2 t + b u 2 ) + 1 p + 1 u p+1 dx = E(t) ct m 1, as claimed. 18
Now we must address that the weights satisfiy the eight conditions. That φ and θ satisfy (i), (ii), and (vi)-(viii) is shown in Radu, Todorova, and Yordanov [2009], while condition (iii) is proven in Todorova and Yordanov [2007]. This leaves (iv) and (v). The integrals in these remaining conditions relate m to C. We now consider two choices of C, which admit separate values of m for different values of the nonlinear exponent p. We will define these values for m in the next sections, and as we do, we will also ensure the weights satisfy conditions (iv) and (v). 19
Chapter 4 Large p Case Theorem 4.0.15. By choosing C = 0 and by choosing m = µ δ = small δ > 0, the weights φ and θ as in (3.0.1) satisfy conditions (iv) and (v). n α δ, for 2 α β Proof. With C = 0, the integrals in conditions (iv) and (v) are trivially satisfied. Further, recall that â 2 = φ 1 (φ tt div(b φ) + aφ t ) as defined in (2.0.2) ( m = + σ(x) ) 2 ( m + t t 2 t 2σ(x) ) ( m + a + σ(x) ) 2 t 3 t t ( 2 b σ(x) 2 div(b σ(x)) ) t 2 t ( div(b σ(x)) am aσ b σ 2 m = + + + σ(x) ) 2 + m t t 2 t t 2 t 2σ 2 t.(4.0.1) 3 To continue, we convert the conditions on A(x) in Conjecture 1.0.1 to conditions on σ(x), as per Radu, Todorova, and Yordanov [2009]: div(b(x) σ(x)) (m + δ)a(x), (4.0.2) 0 < σ(x) (1 + x ) 2 α β, ( 1 δ ) a(x)σ(x) b(x) σ(x) 2. 2µ 20
Further estimating from below, using (4.0.2) in (4.0.1), â 2 ( ) δ (µ δ)a am aσ + aσ 1 2µ + 2σ t t 2 t 3 a(µ m δ) + σ(g 0t γ δ 2µ 2t 1 ) t t 2 a(µ m δ) because 0 γ < 1, and for sufficiently large t. t Thus, for (v) â 2 C = 0 to be true, we require m µ δ. Hence, choosing m = µ δ is sufficient. Similarly, we have that (â 2 ) t 0: ( ( div(b σ(x)) am aσ b σ 2 m (â 2 ) t = + + + σ(x) ) ) 2 + m t t 2 t t 2 t 2σ 2 t 3 t div(b σ(x)) am 2aσ 2b σ 2 = + ( t 2 t 3 m +2 + σ(x) ) ( m + 2σ(x) ) + 2m t t 2 t 2 t 3 t 6σ 3 t ( ) 4 δ (µ δ)a am 2aσ + 2aσ 1 2µ 6σ(m + 1) + t 2 t 3 t 4 a(µ m δ) + σ(g 0t γ δ 6(m + 2µ 1)t 1 ) t 2 t 3 a(µ m δ) 0 because 0 γ < 1 and m = µ δ. t 2 By the prior and (i), â 2 φ (â 2 θ) t = â 2 (φ θ t ) (â 2 ) t θ 0, which shows that the first part of (v) is satisfied. Therefore, with C = 0 and m = n α δ = µ δ, conditions (iv) and (v) hold, and we obtain a powerful energy 2 α β decay estimate. 21
Chapter 5 Small p Case Now we choose a different C that is slightly negative, making â 2 no longer necessarily nonnegative, which allows larger values of m for smaller p. Theorem 5.0.16. There exist some positive constants k and c 1 such that by choosing c 1 t 1 (1 + x ) α if 1 + x kt η, C := (5.0.1) 0, if 1 + x > kt η, where η = 1, and by choosing 2 α β m = 2 p 1 + α p+1 n p 1 2 α β δ, (5.0.2) for small δ > 0, the weights φ and θ as in (3.0.1) satisfy conditions (iv) and (v). Proof. Note that â 2 = div(b σ(x)) am t + ( aσ b σ 2 m + + σ(x) ) 2 + m t 2 t t 2 t 2σ 2 t. 3 22
Using (4.0.2), we get that â 2 (µ δ m)a t + aσ b σ 2 t 2 2σ t 3 a 0 (µ δ m)t 1 (1 + x ) α + δ 2µ t 2 aσ 2t 3 σ = a 0 (µ δ m)t 1 (1 + x ) α + a 0δ 2µ t 2 (1 + x ) 2 2α β 2t 3 σ. Because a g 0 t γ by 1.0.2, δ 2µ is positive, and 0 γ < 1, we have that δ 2µ t 2 aσ will absorb 2t 3 σ, for sufficiently large t. That σ (µ δ)d 0 (1+ x 2 α β was also used. Furthermore, because we want m > µ δ, for t t 0, â 2 c 1 t 1 (1 + x ) α + c 2 t 2 (1 + x ) 2 2α β for some c 1, c 2 > 0. Through an almost identical calculation, since 0 γ < 1, we have, for t t 0, t(â 2 ) t c 3 t 1 (1 + x ) α c 4 t 2 (1 + x ) 2 2α β for some c 3, c 4 > 0. We are now ready to define the slightly negative lower bound of â 2 for small p: c 1 t 1 (1 + x ) α if 1 + x kt η, C := 0, if 1 + x > kt η, {( ) η ( ) η } 1 c1 c3 where η = and k = max,. 2 α β By construction, C satisfies the first part of condition (v), that â 2 C. Condition (iv) requires that â 2 φ (â 2 θ) t k 0 C φ for some positive k 0. Noting that â 2 C c 2 c 4 23
and t(â 2 ) t c 0 C for large t and some c 0 = c3 c1 > 0, we have that, for large t, â 2 φ (â 2 θ) t = â 2 φ t(â 2 ) t t 1 θ â 2 θ t = â 2 (φ θ t ) t(â 2 ) t t 1 θ C (φ θ t + c 0 t 1 θ). Because C 0 and φ > 0, it only remains to be shown that θ t + c 0 t 1 θ cφ for some c > 0. Consider Hence, c 0 t 1 θ c 0 φ/8. Now consider ( (6 θ t = 3 4 t + σ(x) ) 1 φ(x, t)) t 2 t ( (6 = 3 4 t + σ(x) ) 2 ( 6 t 2 t + 2σ(x) 2 t 3 3 ( 6 4 t + σ(x) ) 1 φ t 2 t (x, t) = 3 4 θφ 1 = 3 ( 6 4 t + σ(x) ) 1 φφ 1 (5.0.3) t 2 = 3t ( 6 + σ(x) ) 1 4 t 3t 4 6 1 = t 8. ) ( 6 φ(x, t) + Finally, θ t + c 0 t 1 θ cφ with c = m 8 + c 0 8, which means t + σ(x) ) 1 φ t 2 t (x, t)) ( 6 t + σ(x) ) 1 ( m + σ(x) t 2 t t 2 ) φ mφ 8. â 2 φ (â 2 θ) t C (φ θ t + c 0 t 1 θ) (c + 1)C φ = k 0 C φ where k 0 = c + 1. 24
Lastly, the integral inequalities of (iv) and (v) must be satisfied. To verify them, consider φ 1. Recall, per (4.0.2), σ(x) (1 + x ) 2 α β (kt) η(2 α β) on supp(c ) = (kt) (2 α β)/(2 α β) = kt on supp(c ). Thus, on supp(c ), φ 1 = t m e σ(x) t t m e k (5.0.4) ct m for some positive c. Using (5.0.3), (5.0.4), and the compact support of C, sup t t 0 θφ 2 C p+1 p 1 dx sup ct m+1 t t 0 sup ct m+1 p+1 p 1 t t 0 sup ct m+1 p+1 t t 0 C p+1 p 1 dx kt η 0 B(0,s) p 1 + α p+1 p 1 +n 2 α β, p+1 α s p 1 dσn 1 ds which must be finite to satisfy (v). Therefore, we must have m + 1 p + 1 p 1 α p+1 n p 1 2 α β = m 2 p 1 α p+1 n p 1 2 α β 0. The integral in condition (iv) is even easier to verify and gives another restriction on m that is essentially the same: m p + 1 p 1 α p+1 n p 1 2 α β < 1. (5.0.5) 25
This inequality is strict because there is an integral from t 0 to rather that a supremum over t t 0. Hence, if m satisfies this inequality, both conditions (iv) and (v) are true, and setting m = 2 p 1 + α p+1 n p 1 2 α β δ, for δ small, gives a powerful energy decay estimate for this choice of C. 26
Chapter 6 Combined Results In the previous two chapters, we derived two separate estimates of the energy decay. We now determine which estimates give faster decay as the nonlinear expononent p varies. Proof of Theorem 1.0.3. To begin, recall from Theorem 3.0.14 that E(t) ct m 1. Thus, as m increases, the decay becomes faster. Define m 0 = 2 p 1 δ, m 1 = 2 m 2 = + α p+1 p 1 n p 1 2 α β δ, n α δ. 2 α β From (5.0.2), notice that if we have α p+1 n 0, then p 1 m 2 p 1 + α p+1 n p 1 2 α β δ 2 p 1 δ = m 0, 27
and choosing m = m 0 still satisfies (5.0.5). Thus, a transition between decay rates m 0 and m 1 occurs when α p+1 n = 0. Solving for p gives the first threshold, p 1 p 1 = n + α n α. Furthermore, comparing the two decay rates derived in the previous two chapters gives a second threshold. As shown in Theorem 4.0.15, when C = 0, Setting this equal to m 1 gives m = µ δ = n α 2 α β δ. µ = n α 2 α β = 2 p 1 + α p+1 n p 1 2 α β = m 1 + δ. Thus, we obtain the second threshold for p, p 2 = 1 + 2 β n α. Combining all the previous, we obtain the optimal value of m, a function of p as follows: 2 p 1 δ m = 2 + α p+1 p 1 n p 1 2 α β 2 β if 1 < p 1 +, n α 2 β δ if 1 + < p n+α, n α n α n α δ if n+α < p < n+2, 2 α β n α n 2 which completes the proof of Theorem 1.0.3. 28
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Vita Michael Roberts was born in Knoxville, TN, as the youngest of the five children of Roland and Cathy Roberts. He attended Cedar Bluff Schools and then Farragut High School, where he learned his true love of math. After graduation, he metriculated to the University of Tennessee and earned a Bachelor s degree in Mathematics with a minor in Physics. Passionate about the subjects, he continued on to learn and study partial differential equations. In 2012, he earned his Master of Science in Mathematics at the University of Tennessee. 32