Research Article On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition

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1 Hindawi Publishing Corporation Abstract and Applied Analysis Volume 21, Article ID 68572, 12 pages doi:1.1155/21/68572 Research Article On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition Zhilei Liang School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 61113, China Correspondence should be addressed to Zhilei Liang, zhilei592@gmail.com Received 9 June 21; Accepted 15 September 21 Academic Editor: Nicholas Alikakos Copyright q 21 Zhilei Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We identify the blow-up set of solutions to the problem u t u x p 2 u x x, x >, t,t, u x p 2 u x,t u p 1,t, t,t, andu x, u x, x>, where p>2. We obtain that the blow up set B u satisfies,p p 1 / B u,p p 1 /. The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods. 1. Introduction Consider a one-dimensional process of diffusion in a medium that occupies the half space {x : x }; thatis, u t ( u x p 2 u x x, u x p 2 u x,t u q,t, x >, t,t, t,t, 1.1 u x, u x, x >, where p>2, q>andu x is an appropriately smooth function with some compatibility conditions. Problem 1.1 describes the non-newtonian fluid with a power dependence of the tangential stress on the velocity of the displacement under nonlinear condition. It has many applications and has been intensively studied; see 1 3 and the references cited therein. For the local in time existence, we refer to 4. Also it is known that 1.1 has no classical solution in general due to the possible degeneration at u x. So we usually understand the weak solution defined in the following sense.

2 2 Abstract and Applied Analysis Definition 1.1. A nonnegative function u C,,t with u x L,,t is said to be a weak solution of 1.1, if the integral identity (uφ t u x p 2 u x φ x dx ds u x φ x, dx u q,s φ,s ds 1.2 is fulfilled for all φ C,,t. An interesting phenomenon is that, due to the boundary effect, the solution of 1.1 may exist for t,t and becomes unbounded as t T for some T <. Namely, the solutionoccurs blow-up phenomenon. In this connection, Galaktionov and Levine proved in 5 that the solutions are global in time when <q 2 p 1 /p but occur blow-up for the range 2 p 1 /p < q 2 p 1, while for q>2 p 1 blow-up happens or not depending on the size of the initial data. The main concern in this work is on the set of points at which solutions becomes unbounded, that is, the blow-up set, which is defined as B u {x : x n x, t n T, such that u x n,t n as n }. 1.3 A problem which has attracted a lot of attention in the literature is the identification of possible blow-up sets. Numerical analysis hints that the blow-up set should be a single point single point blow-up when q>p 1, a proper subset of the spatial domain regional blowup when q p 1, and the whole half line global blow-up when <q<p 1. In fact, based on the Gilding and Herrero s work in 6, Quirós and Rossi considered the porous medium type equation u t u m xx, x >, t,t, u m x,t u q,t, t,t, 1.4 u x, u x, x >. They proved in 1 : the blow-up set B u {} if q>mbut B u, if q<m,however, the blow-up set is regional in case of q m, namely, <B u <. Afterwards, Cortázar et al. given a detailed description on the regional blow-up set. They proved in 7 that if q m, then the blow-up set satisfies, 2m/m 1 B u, 2m/m 1. In the light of previous works, we discuss the blow-up set of solutions for the p- Laplacian equation 1.1. In the current paper, we identify the set of blow-up points in case of q p 1. So, in the following we consider u t ( u x p 2 u x x, u x p 2 u x,t u p 1,t, x >, t,t, t,t, 1.5 u x, u x, x >, where p>2. We take u x to be a C 1, nonincreasing and compactly supported function with some compatibility conditions. The below theorem is our main result.

3 Abstract and Applied Analysis 3 Theorem 1.2. Assume that u x C 1, is a nonnegative, nonincreasing and compactly supported function, then all the nontrivial solutions u x, t of problem 1.5 occur blow-up; moreover, the blow-up set B u satisfies [, p( p 1 B u [, p( p 1 ]. 1.6 Remark 1.3. The nonincreasing assumption on u makes the proof much simpler see also 7. Remark 1.4. One could expect that if the solutions of 1.1 occur blow-up, then the blow-up set B u, if <q<p 1, but B u {} if q>p Proof of Theorem 1.2 In order to study the solution u of 1.5 near the blow-up time T, as in 8 we introduce the rescaled function v x, τ T t 1/ p 2 u x, t, 2.1 where τ ln T t /T,. Hence, we obtain the below equations in terms of v by substituting 2.1 into 1.5, v ( τ v x p 2 v x 1 v, x >, τ >, x v x p 2 v x,τ v p 1,τ, τ >, 2.2 v x, T 1/ p 2 u x, x >. If v is τ independent, then u is called a self-similar solution. Let w x T t 1/ p 2 u x, t, we have ( w x p 2 w x x 1 w, x >, 2.3 w x p 2 w x w p 1. A direct integration shows that the non-negative solution of 2.3 is w or ( p 1 / p 2 ( 1/ p 2 ( ( p/ p 2 1 p p 1 w x p 2 ( p 1 x, 2.4 where a max{,a}. Theorem 1.2 is a direct consequence of the following two propositions.

4 4 Abstract and Applied Analysis Proposition 2.1. Let v x, τ be a solution of 2.2, then there exists a function w x, such that the limit holds uniformly on,, v,τ w, as τ. 2.5 Moreover, w x has the explicit form 2.4 and solves problem 2.3. Proposition 2.1 and the formula 2.1 generate [, p( p 1 B u. 2.6 Proposition 2.2. Suppose that the constants a, b satisfy p p 1 / <a<b<, then there exists a large time point τ such that for all τ>τ v x, τ Ce τ/ p 2, x a, b, 2.7 where constant C depends on a, b and u. Proposition 2.2, along with 2.1, shows that u x, t is uniformly bounded for all t,t if x>p p 1 /. This claims B u [, p( p 1 ]. 2.8 Hence, Theorem 1.2 immediately follows from 2.6 and 2.8. So the remaining task in this paper is to prove the validity of Propositions 2.1 and 2.2. This will be discussed in Section Proof of Propositions 2.1 and 2.2 In this section we prove Propositions 2.1 and 2.2. The argument contains several lemmas. Lemma 3.1. All the nontrivial solutions of 1.5 occur blow-up. The proof is available in 5, see also 3. Lemma 3.2. If the initial function u is nonincreasing, then so does the solution u x, t for all existent times, that is, u x, a.e. x, t,,t.

5 Abstract and Applied Analysis 5 Proof. Consider the regularized problem ( u ( p 2 /2ux t u 2 x ɛ, x >, t,t, x ( p 2 /2ux u 2 x ɛ,t u p 1,t, t,t, 3.1 u x, u ɛ x, x >, where ɛ >,u ɛ x C, is nonincreasing and converges to u x uniformly as ɛ. By uniqueness, the solution of 1.5 should be the limit function of that of 3.1. Setting z x, t u 2 x ɛ p 2 /2 u x and then differentiating the equation in t to obtain z t ( u 2 x ɛ p 2 /2uxt ( ( u 2 x ɛ p 2 /2 1u 2 x u xt [ ( p 2 /2 u 2 ( x ɛ ( p 2 /2 1u ] u 2 2 x ɛ x z xx. 3.2 Clearly, 3.2 is a linear parabolic equation with respect to z. On account of the boundary conditions z,t u,t p 1 andz x, u 2 ɛx ɛ p 2 /2 u ɛx, we deduce z x, t via comparison theorem. The proof is completed. Remark 3.3. Using the maximum principle for problem 3.1, weobtain ( p 2 /2ux } { p 2 /2uɛx } max x, t T { u 2 x ɛ max u p 1,t, (u 2 ɛx ɛ. 3.3 By sending ɛ, it follows that { } } max x, t T u x p 2 u x max {u p 1,t, u x p 2 u x. 3.4 Lemma 3.4. There exists a constant C C p, u such that u p 1,s ds C T t 1/ p 2, t,t. 3.5 Proof. Integrating 1.5 1, the first equation of 1.5, over,,t yields u x, t dx u dx u p 1,s ds. 3.6

6 6 Abstract and Applied Analysis Now multiplying by x and integrating the resulting expression once more, we have xu x, t dx xu dx x ( u x p 2 u x dx ds x (x u x p 2 u x x u x p 2 u x dx ds. dx ds u x p 2 u x dx ds 3.7 Thanks to Remark 3.3, we estimate u x p 2 u x dx ds u x,s p 2 u x dx ds u x,s p 2 u,s ds C T, u x u p 1,s ds C T, u x, u x p the last sign comes from the boundary condition. Thus, 3.7 becomes xu x, t dx xu dx u p 1,s ds C T, u x. 3.9 Multiplying 3.6 by a constant K>1 and then subtracting 3.9 produces K x u x, t dx K x u dx K 1 u p 1,s ds C T, u x. 3.1 If we choose K>1 so large such that supp u,k, then K K x u x, t dx K x u dx K x u x, t dx K 2 u,t Inequalities 3.1 and 3.11 lead to K 2 u,t K 1 u p 1,s ds C T, u x Putting F t up 1,s ds and integrating above inequality over t, T brings F t C T t 1/ p 2 C T, u x C 1 T t 1/ p 2, t close to T. 3.13

7 Abstract and Applied Analysis 7 Because u is bounded when time t is away from the blow-up time T, 3.13 and thus 3.5 is valid for all t,t. The next lemma proves that u x, t is localized. Namely, the support of u,t is uniformly bounded for all t,t. Lemma 3.5. There exists a constant M depending on p and u such that if x>m,then u x, t, t,t Proof. Let us introduce the function q x, t D T t 1/ p 2 L x p/ p 2, 3.15 where D satisfies where D p 2 1/2 p 1 p 2/p p 1 and L is an arbitrary positive constant. It is easy to check that q x, t satisfies By differentiating 3.5, we have u,t C T t 1/p 2. This, together with the monotonicity of u, deduces u x, t C T t 1/ p 2, x, t,t Because the initial support is bounded, we choose a suitably large point x < to guarantee supp u,x. Setting ũ x, t u x x,t, it has ũ x, u x x, q x,, x > By virtue of 3.16, there exists some large constant L such that for all L>L,itholds ũ,t u x,t C T t 1/ p DL p/ p 1 T t 1/ p 2 q,t, t,t. So comparison theorem concludes ũ x, t q x, t, x, t,t Hence, for every constant L satisfying L > L, 3.19 ensures supp ũ,t,l. Thus supp u,t,x L. Denoted by M x L, it follows that for all x>m, u x, t, t,t. 3.2 This completes the proof.

8 8 Abstract and Applied Analysis Corollary 3.6. Lemma 3.5 and 2.1 claim that v x, τ is localized as well. Moreover, 2.1 and 3.16 guarantee the boundedness of v x, τ for all x and τ. By a similar argument as that in Lemma 3.2, one can prove that v x p 2 v x also is bounded In what follows, we analyze the large time behavior of the solution v x, τ of problem Multiplying 2.2 by v and then integrating by parts, we deduce that τ2 τ 1 v 2 τdx ds J v τ 2 J v τ 1, a.e., τ 1 <τ 2 <, 3.21 where J v τ is the Liapunov functional and J v τ 1 p v x p dx 1 p v p,τ 1 2 ( v 2 x, τ dx Based on the estimates v, v x L,,, v τ L 2,,, 3.23 the ω-limit set ω v { w x : τ n, s.t. v,τ n w x in L 2} 3.24 consists of solutions of the problem 2.3. Indeed, using the estimates 3.23 in passing to the limit in 2.2, we have that, given a monotone sequence τ n, v,τ n s w x; s in L 2, where w x; s depending on s solves 2.3 in a weak sense, and w x; ω v. It follows from 3.23 that uniformly in s, 1 v,τ n s v,τ n τn s τ n v t 2 L 2 dτ τ n This means that the limit function w does not rely on s and is a weak solution for the 2.3. Finally, the independence of the choice of the sequence τ n follows from the nonincreasing of J v τ in time. Furthermore, the convergence is uniform in x due to the boundness of v x. In conclusion we arrive to the following lemma. Lemma 3.7. The limit holds as τ v x, τ w x, uniformly for x, 3.26 where w x satisfies the problem 2.3. To finish Proposition 2.1, it remains to confirm that w x is not null and thus to be of the form 2.4. The proof will be given at the end of the paper. Now let us turn to Proposition 2.2.

9 Abstract and Applied Analysis 9 Consider the initial-boundary-value problem h τ ( h x p 2 h x x 1 h, x a δ, b δ, τ >, h a δ, τ h b δ, τ ɛ, τ >, 3.27 h x, ɛ, x a δ, b δ, where a, b, ɛ, δ are positive constants and ɛ, δ are suitably small. We have the following. Lemma 3.8. Let Dδ p/ p 2 ɛ suitably small with D that satisfies D p 2 1/2 p 1 p 1 /p p 1. Then there exists a constant C> such that for all τ, h x, τ Ce τ/ p 2, x a, b Proof. It is easy to check that the function of the form D a x p/ p 2, x [ a δ, a δ, 2 j x, x [ a δ 2,b δ ], D x b p/ p 2, x (b δ2,b δ ]. solves ( jx p 2 jx x 1 j, x a δ, b δ, j a δ j b δ Dδ p/ p Moreover, it has j x Dδ p/ p 2 since Dδ p/ p 2 is a super solution to problem 3.3. Choosing Dδ p/ p 2 ɛ and subtracting 3.3 from 3.27, weobtain ( h τ x p 2 h x jx p 2 jx x 1 ( h j, x a δ, b δ, τ >, h a δ, τ j a δ h b δ, τ j b δ, τ >, 3.31 h x, j x, x a δ, b δ.

10 1 Abstract and Applied Analysis Integrating 3.31 over a δ, b δ,τ gives rise to b δ ( b δ ( h j x, τ dx h j x, dx τ τ 1 ( h x p 2 h x jx p 2 jx b δ, s ds ( h x p 2 h x jx p 2 jx a δ, s ds τ b δ x, s dx ds Noting that h j x, τ is symmetric with respect to a b /2 and nonincreasing on a b/2,b δ, thus we have for all τ>, sign {( h x p 2 h x } jx p 2 jx b δ, τ sign {( } h x j x b δ, τ, 3.33 where sign is the sign function. Then, τ ( h x p 2 h x jx p 2 j x b δ, s ds Similarly, τ ( h x p 2 h x jx p 2 jx a δ, s ds Therefore, 3.32 becomes b δ x, τ dx b δ ( 1 τ h j x, dx b δ x, s dx ds Applying the Gronwall inequality yields b δ x, τ dx Ce 1/ p 2 τ. 3.37

11 Abstract and Applied Analysis 11 On the other hand, since h is symmetric and nondecreasing on a b /2,b δ with x- variable, for all x a, b it has b δ/2 h x, τ 1 h ( y, τ dy 1 δ /2 δ 1 δ b δ/2 /2 b δ ( y, τ dy x, τ dx C1 e 1/ p 2 τ, 3.38 where we have used the fact supp j a δ, a δ/2 b δ/2,b δ. That is, h x, τ C 1 e 1/ p 2 τ This completes Lemma 3.8. We are now in a position to prove Proposition 2.2. From Lemma 3.7 we observe that v tends to the limit function w x which is none for all points x>p p 1 /. Taking a>p p 1 / and a small constant δ>tosatisfya δ>p p 1 /. Then, for agivenɛ>, the inequality v x, τ ɛ for all x a δ holds provided τ>τ, where τ is a large point which depends on ɛ. Using comparison theorem, one has v x, τ τ h x, τ, for τ>τ,x>a δ. 3.4 The proof ends up with 3.4 and Lemma 3.8. Finally, We prove that the function w x appeared in Lemma 3.7 takes the form 2.4. For this purpose, it is enough to illuminate that w x is impossible. We argue by contradiction. Assuming that w x for a moment. Then u x, t blows up only at the boundary point x. By this we can find a sequence t n as n such that u,t n as n. Moreover, By the maximin principle see Remark 3.3, one has } { } max x { u x p 2 u x x, t n max u x p 2 u x,u p 1,t n u p 1,t n, 3.41 as long as n is chosen large enough. Recalling that u is nonincreasing with respect to x- variable, from 3.41 we conclude u x x, t n u,t n, x Integrating 3.42 gives rise to 1 x u,t n u x, t n This shows that u occurs blow-up at least in the interval, 1, which contradicts with the assumption that blow-up happens only at x. Now the proof of Theorem 1.2 is completed.

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