Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations

Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations R. Benítez 1,, V. J. Bolós 2, 1 Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda. Virgen del Puerto 2, 16 Plasencia (Cáceres), Spain. e-mail: rbenitez@unex.es 2 Dpto. Matemáticas para la Economía la Empresa, Facultad de Economía, Universidad de Valencia. Avda. Tarongers s/n, 4622 Valencia, Spain. e-mail: vicente.bolos@uv.es Ma 211 (Revised: December 214) Abstract In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, we introduce the concept of blow-up collocation solution and analze numericall some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finall, we discuss the relationships between necessar conditions for blow-up of collocation solutions and exact solutions. 1 Introduction Some engineering and industrial problems are described b explosive phenomena which are modeled b nonlinear integral equations whose solutions exhibit blow-up at finite time (see [1, 2] and references therein). Man authors have studied necessar and sufficient conditions for the existence of a such blow-up time. Particularl, in [3 9], equation (t) = t K (t, s) G ( (s)) ds, t I = [, ˆt [ (1) was considered. In these works, conditions for the existence of a finite blow-up time, as well as upper and lower estimates of it, were given, although the were not ver accurate in some cases. A wa for improving these estimations is to stud numerical approximations of the solution; in this aspect, collocation methods have proven to be a ver suitable technique for approximating nonlinear integral equations, because of its stabilit and accurac (see [1]). Hence, the aim of this paper is to test the usefulness of collocation methods for detecting blow-up solutions of the nonlinear homogeneous Volterra-Hammerstein integral equation (HVHIE) given b equation (1). Work supported b project MTM28-546, Spain. 1

Ver recentl, Yang and Brunner [11] analzed the blow-up behavior of collocation solutions for a similar Hammerstein-Volterra integral equation with a convolution kernel (t) = φ(t) + t K(t s)g(s, (s))ds, t I. (2) Equation (1) should not be regarded as a particular case of (2) (for a convolution kernel), because in [11] a positive non-homogeneit φ and a Lipschitz-continuous nonlinearit G were considered. Indeed, under the hpotheses on the kernel and the nonlinearit considered in [11], it is well-known that equation (1) has onl the trivial solution ( ) and moreover, under the conditions we shall state in Section 3, the collocation equations given in [11] lead to the trivial sequence ( n = for all n N). Thus, equation (1) considered here is beond the scope of the standard techniques, which usuall impose conditions that guarantee the uniqueness of the solutions. An approach for solving this problem, which we shall follow in this paper, is writing equation (1) as an implicitl linear homogeneous Volterra integral equation (HVIE), i.e. setting z = G and plugging it into the nonlinear integral equation (1) to obtain ( t ) z(t) = G ((Vz) (t)) = G K (t, s) z(s) ds, t I, (3) being V the linear Volterra operator. There is a one-to-one correspondence between solutions of (1) and (3) (see [1, 12]), in particular, if z is a solution of (3), then := Vz is a solution of (1). This paper is structured as follows: in Section 2 we introduce briefl the basic definitions and state the notation. Next, we devote Section 3 to generalize the results about existence and uniqueness of nontrivial collocation solutions obtained in [13] in order to appl the results to blow-up problems. In Section 4, we introduce the concept of blow-up collocation solution and analze numericall some blow-up time estimates using collocation methods in particular examples where the previous results about existence and uniqueness can be applied. Finall, in Section 5, we discuss the relationships between necessar conditions for blow-up of collocation solutions and exact solutions. 2 Collocation problems for implicitl linear HVIEs Following the notation of [1], a collocation problem for equation (3) in I = [, T ] is given b a mesh I h := {t n : = t < t 1 <... < t N = T } and a set of m collocation parameters c 1 <... < c m 1. We denote h n := t n+1 t n, n =,..., N 1, and the quantit h := max {h n : n =,... N 1} is the diameter of I h. Moreover, the collocation points are given b t n,i := t n + c i h n, i = 1,..., m, and the set of collocation points is denoted b X h (see [1, 15]). A collocation solution z h is then given b the collocation equation ( t ) z h (t) = G K (t, s) z h (s) ds, t X h, where z h is in the space of piecewise polnomials of degree less than m. From now on, a collocation problem or a collocation solution will be alwas referred to the implicitl linear HVIE (3). So, if we want to obtain an estimation of a solution of the nonlinear HVHIE (1), then we have to consider h := Vz h. At the beginning of Section 4, we define the concepts of blow-up collocation problems and blow-up collocation solutions. To do this, we extend the definition of collocation In [1, 13], the diameter h is also called stepsize. Here, in order to emphasize the variabilit of the stepsizes, we shall not follow this notation for preventing misunderstandings. Moreover, in the algorithms presented in Section 4.1, the initial stepsize h is set equal to h, but in successive iterations the stepsizes ma decrease. 2

solution to meshes I h with infinite points. Taking this into account, it is noteworth that if z h is a blow-up collocation solution, then h also blows up at the same blow-up time. As it is stated in [1], a collocation solution z h is completel determined b the coefficients Z n,i := z h (t n,i ), for n =,..., N 1 and i = 1,..., m, since z h (t n + vh n ) = m j=1 L j (v) Z n,j for all v ], 1], where L 1 (v) := 1, L j (v) := m v c k k j c j c k, for j = 2,..., m, are the Lagrange fundamental polnomials with respect to the collocation parameters. The values of Z n,i are given b the sstem where and Z n,i = G F n (t n,i ) + h n B n (i, j) := ci F n (t) := The term F n (t n,i ) is called the lag term. m j=1 B n (i, j) Z n,j, (4) K (t n,i, t n + sh n ) L j (s) ds tn K (t, s) z h (s) ds. (5) 3 Existence and uniqueness of nontrivial collocation solutions In [13], we used collocation methods for approximating the nontrivial solutions of (1). In particular, we studied the collocation solutions of the implicitl linear HVIE (3), under the assumption that some general conditions were held. Specificall, it was required that the kernel K was a locall bounded function; nevertheless, non locall bounded kernels, e.g. weakl singular kernels, appear in man cases of blow-up solutions (see [7, 14]) and so, in the present work, we shall first replace this condition with another not excluding this kind of kernels. Hence, the general conditions that we are going to impose (even if the are not explicitl mentioned) are: Over K. The kernel K : R 2 [, + [ has its support in { (t, s) R 2 : s t }. For ever t >, the map s K (t, s) is locall integrable, and K(t) := t K (t, s) ds t is a strictl increasing function. Moreover, lim t a + a K (t, s) ds = for all a. Note that for convolution kernels (i.e. K(t, s) = k(t s)) this last condition is alwas held, because k is locall integrable and then lim t a + t a k(t s) ds = lim t a + t a k(s) ds = lim t + t k(s) ds =. Over G. The nonlinearit G : [, + [ [, + [ is a continuous, strictl increasing function, and G() =. Note that, since G is injective, the solution of equation (1) is also given b = G 1 z, where z is a solution of (3). Also, since G() =, the zero function is alwas a solution of both equations (1) and (3), called trivial solution. Moreover, for convolution kernels, given a solution z(t) of (3), an horizontal translation of z, z c (t) defined b {, if t < c z c (t) = z(t c) if t c, This last condition is new with respect to those given in [13], and replaces the locall boundedness condition. 3

is also a solution of (3) (see [13,16,17]). Motivated b this fact we give the next definitions, that are also valid for nonconvolution kernels: Definition 3.1. We sa that a propert P holds near zero if there exists ɛ > such that P holds on ], δ[ for all < δ < ɛ. On the other hand, we sa that P holds awa from zero if there exists τ > such that P holds on ]t, + [ for all t > τ. The concept awa from zero was originall defined in a more restrictive wa in [13]; nevertheless both definitions are appropriate for our purposes. Definition 3.2. We sa that a solution of (3) is nontrivial if it is not identicall zero near zero. Moreover, given a collocation problem, we sa that a collocation solution is nontrivial if it is not identicall zero in ]t, t 1 ]. Given a kernel K, a nonlinearit G and some collocation parameters {c 1,..., c m }, in [13] there were defined three kinds of existence of nontrivial collocation solutions (of the corresponding collocation problem) in an interval I = [, T ] using a mesh I h : existence near zero, existence for fine meshes, and unconditional existence. For these two last kinds of existence, it is ensured the existence of nontrivial collocation solutions in an interval, and hence, there is no blow-up. On the other hand, existence near zero is equivalent to the existence of a nontrivial collocation solution with adaptive stepsize, as it is defined in [11]. In this case, collocation solutions can alwas be extended a little more, but it is not ensured the existence of nontrivial collocation solutions for arbitrar large T and, thus, it can blow up in finite time. For the sake of clarit and readabilit, let us recall the definition of existence near zero given in [13]: Definition 3.3. We sa that there is existence near zero if there exists H > such that if < h H then there are nontrivial collocation solutions in [, t 1 ]; moreover, there exists H n > such that if < h n H n then there are nontrivial collocation solutions in [, t n+1 ] (for n = 1,..., N 1 and given h,..., h n 1 > such that there are nontrivial collocation solutions in [, t n ]). Note that, in general, H n depends on h,..., h n 1. As in [13], we shall restrict our analsis to two particular cases of collocation problems, namel: Case 1: m = 1 with c 1 >. Case 2: m = 2 with c 1 =. In these cases, sstem (4) is reduced to a single nonlinear equation, whose solution is given b the fixed points of G (α + β) for some α, β. Determining conditions for the existence of collocation solutions in the general case is a problem of an overwhelming difficult. In fact, in [13] examples of equations for which, in cases 1 and 2, there existed nontrivial collocation solutions, et in some other cases, for the same equation, there were no nontrivial collocation solutions, were given. The proofs of the results obtained in [13] rel on the locall boundedness of the kernel K. Therefore, we will revise such proofs, imposing the new general conditions given at the beginning of this section, and avoiding other hpotheses that turn out to be unnecessar. For example, we extend these results to decreasing convolution kernels, among others. From the previous work [13], we quote a lemma on nonlinearities G satisfing the general conditions: Lemma 3.1. The following statements are equivalent to the statement that G() (in ], + [): is unbounded (i) There exists β > such that G (β) has nonzero fixed points for all < β β. (ii) Given A, there exists β A > such that G (α + β) has nonzero fixed points for all α A and for all < β β A. 4

3.1 Case 1: m = 1 with c 1 > First, we shall consider m = 1 with c 1 >. Equations (4) are then reduced to where Z n,1 = G (F n (t n,1 ) + h n B n Z n,1 ), n =,..., N 1, (6) B n := B n (1, 1) = c1 K (t n,1, t n + sh n ) ds (7) and the lag terms F n (t n,1 ) are given b (5) with i = 1. Note that B n > because the integrand in (7) is strictl positive almost everwhere in ], c 1 [. Remark 3.1. In [13], it is ensured that lim hn + h nb n = taking into account (7) and the old general conditions imposed over K. Thus, we must ensure that the limit also vanishes considering the new general conditions. In fact, with a change of variable, we can express (7) as B n = 1 tn,1 K (t n,1, s) ds. (8) h n t n Hence, b (8) and the new condition, we have since t n,1 = t n + c 1 h n. t lim h nb n = lim K (t, s) ds =, h n + t t + n t n Now, we are in position to give a characterization of the existence near zero of nontrivial collocation solutions: Proposition 3.1. There is existence near zero if and onl if G() is unbounded. Proof. ( ) Let us prove that if G() is unbounded, then there is existence near zero. So, we are going to prove b induction over n that there exist H n >, for n =,..., N 1, such that if < h n H n then there exist solutions of the sstem (6) with Z n,1 > : For n =, taking into account Remark 3.1 and Lemma 3.1-(i), we choose a small enough H > such that < h B β for all < h H. So, since the lag term is, we can appl Lemma 3.1-(i) to the equation (6), concluding that there exist strictl positive solutions for Z,1. Let us suppose that, choosing one of those Z,1 and given n >, there exist H 1,..., H n 1 > such that if < h i H i with i = 1,..., n 1, then there exist coefficients Z 1,1,..., Z n 1,1 fulfilling the equation (6). Note that these coefficients are strictl positive, and hence, it is guaranteed that the corresponding collocation solution z h is (strictl) positive in ], t n ]. Finall, we are going to prove that there exists H n > such that if < h n H n then there exists Z n,1 > fulfilling the equation (6) with the previous coefficients Z,1,... Z n 1,1 : Let us define A := max {F n (t n + c 1 ɛ) : ɛ 1}. Note that A exists because s K (t, s) is locall integrable for all t >, and z h is bounded in [, t n ]; hence, F n, given b (5), is bounded in [t n, t n + c 1 ]. Moreover, A because K and z h are positive functions. So, appling Lemma 3.1-(ii), there exists β A > such that G (α + β) has positive fixed points for all α A and for all < β β A. On one hand, taking into 5

account Remark 3.1, we choose a small enough < H n 1 such that < h n B n β A for all < h n H n ; on the other hand, choosing one of those h n, we have F n (t n,1 ) A. Hence, we obtain the existence of Z n,1 as the strictl positive fixed point of G (F n (t n,1 ) + h n B n ). ( ) For proving the other condition, we use Lemma 3.1-(i), taking into account Remark 3.1. A similar result was proved in [13] with the additional hpothesis K (t, s) K (t, s) for all s t < t (or k is increasing for the particular case of convolution kernels K(t, s) = k(t s)). We have shown that this hpothesis is not needed and the difference between the proofs is the choice of A. Taking into account Remark 3.1, we can adapt the results given in [13] about sufficient conditions on existence for fine meshes and unconditional existence to our new general conditions. This will help us to identif collocation problems without blow-up (see Proposition 3.2 below). In the following results recall the Definition 3.1 of the concepts near zero and awa from zero. Proposition 3.2. Let G() be unbounded near zero. If K is a convolution kernel and G() is bounded awa from zero, then there is existence for fine meshes. If G() is unbounded awa from zero, then there is unconditional existence. If, in addition, G() is a strictl decreasing function, then there is at most one nontrivial collocation solution. The proof is analogous to the one given in [13]. Note that the propert G() is unbounded awa from zero appears there as there exists a sequence { n } + n=1 of positive real numbers G( and divergent to + such that lim n) n + n =, but both are equivalent. 3.2 Case 2: m = 2 with c 1 = Considering m = 2 with c 1 =, we have to solve the following equations: for n =,..., N 1, where Z n,1 = G (F n (t n,1 )) (9) Z n,2 = G (F n (t n,2 ) + h n B n (2, 1) Z n,1 + h n B n (2, 2) Z n,2 ), (1) B n (2, j) = c2 K (t n,2, t n + sh n ) L j (s) ds, j = 1, 2 (11) and F n (t n,i ), for i = 1, 2, are given b (5). Note that B n (2, j) > for j = 1, 2, because the integrand in (11) is strictl positive almost everwhere in ], c 2 [. Remark 3.2. As in the first case, we have to ensure that lim hn + h nb n (2, j) = for j = 1, 2, taking into account the new general conditions (see Remark 3.1). Actuall, since < L j (s) < 1 for s ], c 2 [, we have c2 h n B n (2, j) h n K (t n,2, t n + sh n ) ds = tn,2 t n K (t n,2, s) ds. (12) Hence, b (12) and the general conditions, the following inequalities are fulfilled: lim h n + h n B n (2, j) lim t t + n t t n K (t, s) ds =. 6

Analogousl to the previous case, we present a characterization of the existence near zero of nontrivial collocation solutions: Proposition 3.3. Let the map t K (t, s) be continuous in ]s, ɛ[ for some ɛ > and for all s < ɛ (this hpothesis can be removed if c 2 = 1). Then there is existence near zero if and onl if G() is unbounded. Proof. ( ) Let us prove that if G() is unbounded, then there is existence near zero. So, we are going to prove b induction over n that there exist H n >, for n =,..., N 1, such that if < h n H n then there exist solutions of the sstem (1) with Z n,2 > : For n =, taking into account Remark 3.2 and Lemma 3.1-(i), we choose a small enough H > such that < h B (2, 2) β for all < h H. So, since the lag terms are and Z,1 = G () =, we can appl Lemma 3.1-(i) to the equation (1), concluding that there exist strictl positive solutions for Z,2. Let us suppose that, choosing one of those Z,2 and given n >, there exist constants H 1,..., H n 1 > such that if < h i H i, for i = 1,..., n 1, then there exist coefficients Z 1,2,..., Z n 1,2 fulfilling the equation (1). Moreover, let us suppose that z h is positive in [, t n ], i.e. these coefficients satisf Z l,2 (1 c 2 ) Z l,1 for l = 1,..., n 1, where Z l,1 is given b (9). Finall, we are going to prove that there exists H n > such that if < h n H n then there exists Z n,2 > fulfilling the equation (1) with the previous coefficients, and z h is positive in [, t n+1 ], i.e. Z n,2 (1 c 2 ) Z n,1 : Let us define A := max {F n (t n + c 2 ɛ) + (13) c2 } ɛg (F n (t n + c 1 ɛ)) K (t n,2, t n + sh n ) L 1 (s) ds : ɛ 1. Note that A exists and A because s K (t, s) is locall integrable for all t >, z h is bounded and positive in [, t n ], the nonlinearit G is bounded and positive, and the polnomial L 1 is bounded and positive in [, c 2 ]. So, appling Lemma 3.1-(ii), there exists β A > such that G (α + β) has positive fixed points for all α A and for all < β β A. On one hand, taking into account Remark 3.2, we choose a small enough < H n 1 such that < h n B n (2, 2) β A for all < h n H n ; on the other hand, taking into account (5), the lag terms F n (t n,i ) are positive for i = 1, 2, because K and z h are positive. Therefore, b (9), Z n,1 = G (F n (t n,1 )) is positive, because G is positive. Moreover, h n B n (2, 1) is positive. So, F n (t n,2 ) + h n B n (2, 1) Z n,1 A. Hence, we obtain the existence of Z n,2 as the strictl positive fixed point of the function G (α + β) where α := F n (t n,2 ) + h n B n (2, 1) Z n,1 and β := h n B n (2, 2). Concluding, we have to check that Z n,2 (1 c 2 ) Z n,1. On one hand, 2 Z n,2 = G F n (t n,2 ) + h n B n (2, j) Z n,j G (F n (t n,2 )) j=1 because B n (2, j) Z n,j for j = 1, 2. On the other hand, since t K (t, s) is continuous in ]s, ɛ[ for some ɛ > and for all s < ɛ, we can suppose that the stepsize is small enough for t n,1 < ɛ, and then F n (t) is continuous in t n,1. Hence, 7

since G is continuous, G (F n (t)) is also continuous in t n,1, i.e. lim hn + G (F n (t n,2 )) = G (F n (t n,1 )). Therefore, choosing a small enough h n, we have G (F n (t n,2 )) (1 c 2 ) G (F n (t n,1 )) = (1 c 2 ) Z n,1. ( ) For proving the other condition, we use Lemma 3.1-(i), taking into account Remark 3.2. It is noteworth that a similar result was proved in [13], but the hpothesis on the kernel was K (t, s) K (t, s) for all s t < t, and so, unlike Proposition 3.3, it could not be applied to decreasing convolution kernels (for example). Again, the main difference between both proofs lies in the choice of A. Note that for convolution kernels K(t, s) = k(t s), the hpothesis on K is equivalent to sa that k is continuous near zero, which is a fairl weak hpothesis. On the other hand, for general kernels, this hpothesis is onl needed to ensure that t K (t, s) is continuous in t n,1. Therefore, if it is not continuous, it is important to choose h i such that the mapping t K (t, s) is continuous at the collocation points t i,1, i =,... N 1. More specificall, the hpothesis on K is onl needed to check that Z n,2 (1 c 2 ) Z n,1, and hence, to ensure that z h is positive in [, t n ]. Nevertheless, this hpothesis is not needed to verif that Z,2 > Z,1 =, and so, it is alwas ensured the existence of Z 1,2 (and obviousl Z 1,1 ), even if the hpothesis does not hold. Hence, if we use another method to check that Z 1,2 (1 c 2 ) Z 1,1 (e.g., numericall), then it is ensured the existence of Z 2,2. Repeating this reasoning we can guarantee the existence of Z n,2, removing the hpothesis on K. Thus, we can state a result analogous to Proposition 3.3 without hpothesis on the kernel: G() Proposition 3.4. is unbounded if and onl if there exists H > such that there are nontrivial collocation solutions in [, t 1 ] for < h H. In this case, there alwas exists H 1 > such that there are nontrivial collocation solutions in [, t 2 ] for < h 1 H 1. Moreover, if G() is unbounded and there is a positive nontrivial collocation solution in [, t n ], then there exists H n > such that there are nontrivial collocation solutions in [, t n+1 ] for < h n H n. Considering Remark 3.2 and taking the coefficient A given in (13), we can adapt some results of [13] about sufficient conditions on existence for fine meshes and unconditional existence to our new general conditions. These results are useful for identifing collocation problems without blow-up. Proposition 3.5. Let G() be unbounded near zero, and let the map t K (t, s) be continuous in ]s, ɛ[ for some ɛ > and for all s < ɛ (this hpothesis can be removed if c 2 = 1). If K is a convolution kernel and G() is bounded awa from zero, then there is existence for fine meshes. If G() is unbounded awa from zero, then there is unconditional existence. If, in addition, G() is a strictl decreasing function, then there is at most one nontrivial collocation solution. Moreover, as in Proposition 3.4, we can state a result about unconditional existence without hpothesis on the kernel: Proposition 3.6. Let G() be unbounded near zero. If G() is unbounded awa from zero and there is a positive nontrivial collocation solution in [, t n ], then there is a nontrivial collocation solution in [, t n+1 ] for an h n >. If, in addition, G() is a strictl decreasing function, then there is at most one nontrivial collocation solution. 8

3.3 Nondivergent existence and uniqueness Our interest is the stud of existence of nontrivial collocation solutions using meshes I h with arbitrar small h >. So, we are not interested in collocation problems whose collocation solutions escape to + when a certain h n +, since this is a divergence smptom. Following this criterion, we define the concept of nondivergent existence : Let = t <... < t n be a mesh such that there exist nontrivial collocation solutions, and let S hn be the index set of the nontrivial collocation solutions of the corresponding collocation problem with mesh t <... < t n < t n + h n. Given s S hn, we denote b Z s;n,i the coefficients of the corresponding nontrivial collocation solution satisfing equations (4). Then, we sa that there is nondivergent existence in t + n if { } inf s S hn max i=1,...,m {Z s;n,i} exists for small enough h n > and it does not diverge to + when h n +. Given a mesh I h = { = t <... < t N } such that there exist nontrivial collocation solutions, we sa that there is nondivergent existence if there is nondivergent existence in t + n for n =,..., N. See [13] for a more detailed analsis, where we also define the concept of nondivergent uniqueness. In [13] it is proved the next result: Proposition 3.7. In cases 1 and 2 with existence of nontrivial collocation solutions, there is nondivergent existence if and onl if G() is unbounded near zero. Moreover, if, in addition, G is well-behaved, in the sense stated below, then there is nondivergent uniqueness. We sa that G is well-behaved if G(α+) is strictl decreasing near zero for all α >. Note that this condition is ver weak (see [13]). So, taking into account Propositions 3.1, 3.3 and 3.7, the main result of this paper is: Theorem 3.1. (Hpothesis onl for case 2 with c 2 1: the map t K (t, s) is continuous in ]s, ɛ[ for some ɛ > and for all s < ɛ.) There is nondivergent existence near zero if and onl if G() is unbounded near zero. Moreover, if in addition, G is well-behaved then, there is nondivergent uniqueness near zero. In the same wa, we can combine Propositions 3.4 and 3.7 obtaining a result about nondivergent existence and uniqueness without hpothesis on the kernel, and, as Theorem 3.1, it can be useful to stud numericall problems with decreasing convolution kernels in case 2 with c 2 1. We can also combine Propositions 3.2, 3.5 and 3.6 with Proposition 3.7, obtaining results about nondivergent existence and uniqueness (for fine meshes and unconditional) that are useful for identifing collocation problems without blow-up. Some of these results can be reformulated as necessar conditions for the existence of blow-up (see Section 5). 4 Blow-up collocation solutions In this section we will extend the concept of collocation problem and collocation solution in order to consider the case of blow-up collocation solutions. Definition 4.1. We sa that a collocation problem is a blow-up collocation problem (or has a blow-up) if the following conditions are held: 1. There exists T > such that there is no collocation solution in I = [, T ] for an mesh I h. 2. Given M > there exists < τ < T, and a collocation solution z h defined on [, τ] such that z h (t) > M for some t [, τ]. 9

We can not speak about blow-up collocation solutions in the classic sense, since collocation solutions are defined in compact intervals and obviousl the are bounded; so, we have to extend first the concept of collocation solution to half-closed intervals I = [, T [ before we are in position to define the notion of blow-up collocation solution. Definition 4.2. Let I := [, T [ and I h be an infinite mesh given b a strictl increasing sequence {t n } + n= with t = and convergent to T. A collocation solution on I using the mesh I h is a function defined on I such that it is a collocation solution (in the classic sense) for an finite submesh {t n } N n= with N N. A collocation solution on I is a blow-up collocation solution (or has a blow-up) with blow-up time T if it is unbounded. Remark 4.1. In [11] it is defined the concept of collocation solution with adaptive stepsize that uses an infinite mesh I h. It is stated that this kind of collocation solutions blows up in finite time if T b (I h ) := lim t n = lim n n n h i <, without imposing an condition about unboundedness on the collocation solution. In this case, T b (I h ) is called the numerical blow-up time. According to this definition, we could remove the second point in Definition 4.1. However, this definition of numerical blow-up time in [11] is made in the framework of a Lipschitz nonlinearit G in the non-homogeneous case, and it is proved (in case 1) that the corresponding collocation solution is effectivel unbounded if we choose a given adaptive stepsize that depends on the Lipschitz constant of G. Nevertheless, we can not use this framework in our case (homogeneous case), because supposing a Lipschitz-continuous nonlinearit G would impl that the unique solution of (1) or (3) is the trivial one. On the other hand, if we consider a non-lipschitz nonlinearit, then we can not assure that the collocation solution is unbounded since the adaptive stepsize given in [11] becomes zero in the homogeneous case. Therefore, in Definition 4.1 it is necessar the condition about unboundedness, and in Definition 4.2 we impose explicitl that the collocation solution has to be unbounded. Given a collocation problem with nondivergent uniqueness near zero, a necessar condition for the nondivergent collocation solution to blow-up is that there is neither existence for fine meshes nor unconditional existence. So, for example, given a convolution kernel K(t, s) = k(t s), in cases 1 and 2 we must require that G() is unbounded awa from zero (moreover, in case 2 with c 2 1, we must demand that there exists ɛ > such that k is continuous in ], ɛ[). For instance, in [5] it is studied equation (1) with convolution kernel k(t s) = (t s) β, β, and nonlinearit G() = t α, < α < 1, concluding that the nontrivial (exact) solution does not have blow-up. If we consider a collocation problem with the above kernel and nonlinearit in cases 1 and 2, then we can ensure unconditional nondivergent uniqueness (b Propositions 3.2, 3.5 and 3.7); thus, we can also conclude that the nondivergent collocation solution has no blow-up. Actuall, we reach the same conclusion considering an kernel satisfing the general conditions (in case 1 and case 2 with c 2 = 1) and such that t K (t, s) is continuous in ]s, ɛ[ for some ɛ > and for all s < ɛ (in case 2 with c 2 1). In Section 5 we discuss in more detail the relationships between necessar conditions for blow-up of collocation solutions and exact solutions. 4.1 Numerical algorithms Given a blow-up collocation problem with nondivergent uniqueness near zero (alwas in cases 1 and 2), we are going to describe a general algorithm to compute the nontrivial collocation i= 1

solution and estimate the blow-up time. Given stepsizes h,..., h n, the collocation solution at t n+1 is obtained from the attracting fixed point of a certain function G(α+β), given b equation (6) in case 1, or (1) in case 2. Therefore, the ke point in the algorithm is to decide whether there is a fixed point or not and, if so, estimate it. In order to check that there is no fixed point, the most straightforward technique consists on iterating the function and, if a certain bound (which ma depend on the fixed points found in the previous steps) is overcomed, then it is assumed that there is no fixed point. So, if for a certain n there is no fixed point, then a smaller h n should be taken (e.g. h n /2 is used in the examples below), and this procedure is repeated with the new α and β corresponding to this new value of h n. If h n becomes smaller than a given tolerance (e.g. 1 12 in our examples), the algorithm stops and t n = h +... + h n 1 is the estimation of the blow-up time. Finall, we need to determine whether the obtained collocation solution is unbounded or not, but it is not strictl possible since we have computed the collocation solution onl for a finite mesh. Therefore, we will check if the collocation solution overcomes a previousl fixed threshold bound M >. To sum up, the algorithm consists mainl on: 1. Set n = and an initial stepsize h (= h). 2. Check if there is a fixed point in (6) (case 1) or (1) (case 2), taking into account Section 2. 3. If there is fixed point, then compute it, set n = n + 1, h n = h n 1, and repeat 2. 4. If there is not fixed point, then repeat 2 with a smaller stepsize h n. If h n becomes smaller than a given tolerance, the algorithm ends and the blow-up time estimation is given b t n = h +... + h n 1. Note that the initial stepsize h is also denoted b h because it coincides with the diameter of the mesh. 4.2 Examples We will consider the following examples: if [, 1] 1. K (t, s) = 1 and G () = 2 if ]1, + [ if [, 1] 2. K (t, s) = 1 and G () = e 1 if ]1, + [ 3. K (t, s) = k(t s) = t s and G () of Example 1. 1 4. K (t, s) = k(t s) = and G () of Example 1. π(t s) In Example 1, the fixed points of G(α + β) can be found analticall: If α < 1 and 1 α < β 1 4α, then the fixed point is 1 2αβ 1 4αβ ( If α < 1 and β 1 α, then the fixed point is 1 2 β + ) β 2 + 4α. If α 1 and β 1 4α, then the fixed point is 1 2αβ 1 4αβ 2β 2. 2β 2. 11

Figure 1: Extrapolation technique given b (14). M 1, M 2 and m 1, m 2 are the maximum and minimum times respectivel, using initial stepsizes h 1 and h 2, with h 1 > h 2. If β > 1 4α then there is no solution for the corresponding h n. Hence a smaller h n is taken as described in the general case. However, both the computational cost and accurac of the general and specific algorithms are similar. In the first two examples, the blow-up time of the corresponding (exact) solution is ˆt = 3, while in the other two the blow-up time is unknown. In [7] it was studied a famil of equations to which Example 4 belongs. In all examples, both K and G fulfill the general conditions and the hpotheses of Theorem 3.1, and thus it is ensured the nondivergent existence and uniqueness near zero. Note that since the kernel in Example 4 is decreasing and unbounded near zero this case is out of the scope of the stud of [13]. On the other hand, in all examples, G() is unbounded awa from zero and G() is bounded awa from zero, and hence there can exist a blow-up (see Propositions 3.2 and 3.5). We have found the numerical nondivergent collocation solutions for a given diameter h (it is in fact the initial stepsize h ), using the algorithms described in Section 4.1 (the specific for the Example 1 and the general for the rest). In Figures 2, 3, 4 and 5 the estimations of the blow-up time of the collocation solutions for a given initial stepsize h are depicted, varing c 1 in case 1 or c 2 in case 2. Note that the different graphs for different h intersect each other in a fairl good approximation of the blow-up time, as it is shown in Tables 1, 2, 4 and 5. Moreover, we can use a more general technique consisting on extrapolating the minimum (m 1, m 2 ) and maximum (M 1, M 2 ) times of two results with different initial stepsizes h 1, h 2 respectivel, with h 1 > h 2 (see Figure 1); in this wa the blow-up time estimation is given b ˆt M 1 (M 1 M 2 )(M 1 m 1 ) (M 1 M 2 ) + (m 2 m 1 ). (14) In Examples 3 and 4 we do not know the exact value of the blow-up time. However, in order to make a stud of the relative error analogous to the previous examples, we have taken as blow-up time for Example 3 the approximation for h =.1 in case 2 with c 2 =.5: t = 5.78482; in Example 4 the blow-up time value has been taken as the approximation for h =.1 in case 2 with c 2 = 2/3 (Radau I collocation points): t = 1.645842. Results are shown in Tables 7, 8, 1 and 11. The relative error of varing c 1 (case 1) or c 2 (case 2) is the relative vertical size of the graph, and it decreases at the same rate as h. On the other hand, the relative error of the intersection decreases faster, in some cases at the same rate as h 2. Moreover, in case 1, the best approximations are obtained with c 1.5, and in case 2 with c 2 2/3 (approximatel Radau I ) for Examples 1, 2 and 4 (see Tables 3, 6 and 12), while for Example 3 the best approximations are obtained with c 2.5; however, in Table 9 are also shown the approximations and their corresponding errors for the Radau I collocation 12

Figure 2: Example 1. Numerical estimation of the blow-up time of collocation solutions varing c 1 (case 1) or c 2 (case 2), for different initial stepsizes h. The blow-up time of the corresponding (exact) solution is ˆt = 3. The intersection of both curves gives a good approximation of the blow-up time. 13

Figure 3: Example 2. Numerical estimation of the blow-up time of collocation solutions varing c 1 (case 1) or c 2 (case 2), for different initial stepsizes h. The blow-up time of the corresponding (exact) solution is ˆt = 3. The intersection of both curves gives a good approximation of the blow-up time. points. On the other hand, the intersections technique offers better results, but at a greater computational cost. 5 Discussion and comments The main necessar conditions for a collocation problem to have a blow-up are obtained from Propositions 3.2 (case 1) and 3.5 (case 2), and are mostl related to the nonlinearit, since the assumption on the kernel the map t K (t, s) is continuous in ]s, ɛ[ for some ɛ > and for all s < ɛ is onl required in case 2 with c 2 1 and it is a ver weak hpothesis. Hence, assuming that the kernel satisfies this hpothesis, the main necessar condition for the existence of a blow-up is: 1. G() is bounded awa from zero. In addition, for convolution kernels, there is another necessar condition: 2. G() is unbounded awa from zero, i.e. there exists a sequence { n } + n=1 numbers and divergent to + such that lim n n + G( =. n) of positive real In [4] it is given a necessar and sufficient condition for the existence of blow-up (exact) solutions for equation (1) with a kernel of the form K(t, s) = (t s) α 1 r(s) with α >, r 14

Figure 4: Example 3. Numerical estimation of the blow-up time of collocation solutions varing c 1 (case 1) or c 2 (case 2), for different initial stepsizes h. The intersection of both curves gives a good approximation of the blow-up time. 15

Figure 5: Example 4. Numerical estimation of the blow-up time of collocation solutions varing c 1 (case 1) or c 2 (case 2), for different initial stepsizes h. The intersection of both curves gives a good approximation of the blow-up time. 16

nondecreasing and continuous for t, r(t) = for t, and r(t) > for t > : + ( ) 1/α s ds < +. (15) G(s) s This also holds for convolution kernels of Abel tpe K(t, s) = (t s) α 1 with α >, generalizing some results given in [3]. Next we will show that necessar conditions 1 and 2 above mentioned are also necessar conditions for the integral given in (15) to be convergent, and thus for the existence of a blow-up (exact) solution. Proposition 5.1. If (15) holds, then G() is unbounded awa from zero. Proof. Let us suppose that G() that G() < M for all > δ, for a large enough δ. Hence + ( s ) 1/α ds G(s) s > is bounded awa from zero. So, there exists M > such + δ ( ) 1/α ( ) 1/α s ds 1 + G(s) s > ds M δ s = +. Proposition 5.2. If (15) holds, then Proof. B Proposition 5.1, G() increasing sequence { n } + all n. Let us suppose that G() is bounded awa from zero. is unbounded awa from zero and hence, there exists a strictl n=1 with 1 > and divergent to + such that n G( n) < 1 2 α < 1 for G() is unbounded awa from zero; so, we can construct a subsequence of { n } + n=1, that we are going to call {z n} + n=1, such that there exist z n ]z n, z n+1 [ with z n G(z n ) = 1 for each n. This is possible because G() is unbounded awa from zero and then, z given z n there exists z > z n such that G(z) > 1. Then, we choose z n+1 to be an element of { n } + n=1 greater than z. Since z n G(z < 1 and G is continuous, the existence of n) z n is proved. Moreover, since G is positive and strictl increasing, we have that z n G(z > n) and then z n > G (z n ). Hence, taking also into account that z n = G (z n) and have z n G(z n ) = 1, zn G(z < 1 n) 2, we α zn+1 z n ( s ) 1/α ds G(s) s z ( ) n 1/α s ds z ( ) > z n G(s) s > n 1/α s ds z n G (z n) s ( ( ) ) ( 1/α ( ) ) 1/α zn zn = α 1 > α 1 > α G (z n ) 2. z n Therefore, (15) does not hold. The results presented here provide a guide for future research. An interesting problem which is not full resolved is to determine the relationship between the existence of blow-up in exact solutions and in collocation solutions. Another open problem is to assure the unboundedness of a collocation solution with adaptive stepsize (in the sense of [11]) using an infinite mesh I h with lim n t n < in the general case of a non-lipschitz nonlinearit G (see Remark 4.1). In other words, if we can not arrive at a given time T with an collocation solution (i.e. using an mesh), then, is there a collocation blow-up? If so, and in terms of Definition 4.1, it would be sufficient to hold onl the first condition for a collocation problem to have a blow-up. 17

Acknowlegdments The authors would like to thank the anonmous reviewer of the journal Applied Mathematics and Computation for his/her valuable comments that contributed greatl to the improvement of the paper. References [1] C. M. Kirk. Numerical and asmptotic analsis of a localized heat source undergoing periodic motion. Nonlinear Anal. 71 (29), e2168 e2172. [2] F. Calabrò, G. Capobianco. Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs. J. Comput. Appl. Math. 228 (29), 58 588. [3] W. Mdlarczk. A condition for finite blow-up time for a Volterra integral equation. J. Math. Anal. Appl. 181 (1994), 248 253. [4] W. Mdlarczk. The blow-up solutions of integral equations. Colloq. Math. 79 (1999), 147 156. [5] T. Ma lolepsz, W. Okrasiński. Conditions for blow-up of solutions of some nonlinear Volterra integral equations. J. Comput. Appl. Math. 25 (27), 744 75. [6] T. Ma lolepsz, W. Okrasiński. Blow-up conditions for nonlinear Volterra integral equations with power nonlinearit. Appl. Math. Letters. 21 (28), 37 312. [7] T. Ma lolepsz, W. Okrasiński. Blow-up time for solutions for some nonlinear Volterra integral equations. J. Math. Anal. Appl. 366 (21), 372 384. [8] H. Brunner, Z. W. Yang. Blow-up behavior of Hammerstein-tpe Volterra Integral Equations. J. Integral Equations Appl. 24 (4) (212), 487 512. [9] T. Ma lolepsz. Nonlinear Volterra integral equations and the Schröder functional equation. Nonlinear Anal. T. M. A. 74 (211), 424-432. [1] H. Brunner. Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge Universit Press, Cambridge (24). [11] Z. W. Yang, H. Brunner. Blow-up behavior of collocation solutions to Hammerstein-tpe Volterra integral equations. SIAM J. Numer. Anal. 51 (213), 226 2282. [12] M. A. Krasnosel skii, P. P. Zabreiko. Geometric Methods of Nonlinear Analsis. Springer Verlag, New York, 1984. [13] R. Benítez, V. J. Bolós. Existence and uniqueness of nontrivial collocation solutions of implicitl linear homogeneous Volterra integral equations. J. Comput. Appl. Math. 235 (211), 3661 3672. [14] T. Ma lolepsz. Blow-up solutions in one-dimensional diffusion models. Nonlinear Anal. T. M. A. 95 (214), 632-638. [15] H. Brunner. Implicitl linear collocation methods for nonlinear Volterra integral equations. Appl. Numer. Math. 9 (1992), 235 247. [16] M.R. Arias, R. Benítez, Aspects of the behaviour of solutions of nonlinear Abel equations. Nonlinear Anal. T.M.A. 54 (23), pp. 1241 1249. [17] M.R. Arias, R. Benítez, A note of the uniqueness and the attractive behaviour of solutions for nonlinear Volterra equations, J. Integral Equations Appl. 13 (4) (21), 35 31. 18

Tables Case 1 Varing c 1 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 2.6 3.66.1 4 1 1 (c 1 = 1) (c 1 =.1) 3.33 1.1 1 2 3.3 2.78 3.4.5 2 1 1 (c 1 =.39) (c 1 = 1) (c 1 =.1) 1 2 2.92 3.11.1 6 1 2 (c 1 = 1) (c 1 =.1) 3.44 1.5 1 3 3.2 2.96 3.6.5 3 1 2 (c 1 =.44) (c 1 = 1) (c 1 =.1) 7 1 4 Table 1: Example 1. Numerical data of Figure 2 (case 1: m = 1, c 1 > ). Case 2 Varing c 2 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 2.95 3.2.1 8 1 2 (c 2 = 1) (c 2 =.1) 3.8 2.6 1 3 3.3 2.98 3.1.5 4 1 2 (c 2 =.618) (c 2 = 1) (c 2 =.1) 1 4 2.995 3.2.1 8 1 3 (c 2 = 1) (c 2 =.1) 3.8 2.6 1 4 3.3 2.998 3.1.5 4 1 3 (c 2 =.623) (c 2 = 1) (c 2 =.1) 1 6 Table 2: Example 1. Numerical data of Figure 2 (case 2: m = 2, c 1 = ). h Case 1 - c 1 =.5 Case 2 - Radau I Blow-up Rel. error Blow-up Rel. error.1 2.933883 2.2 1 2 2.995253 1.6 1 3.5 2.9652 1.2 1 2 2.99762 8 1 4.1 2.992885 2.4q 1 3 2.999519 1.6 1 4.5 2.996434 1.2 1 3 2.999759 8 1 5 Table 3: Example 1. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with c 1 =.5, and case 2 with Radau I collocation points, c 1 =, c 2 = 2/3, for different initial stepsizes h. 19

Case 1 Varing c 1 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 2.68 3.5.1 3 1 1 (c 1 = 1) (c 1 =.1) 3.18 6 1 3 3.15 2.82 3.3.5 1.5 1 1 (c 1 =.399) (c 1 = 1) (c 1 =.1) 5 1 3 2.94 3.8.1 4.6 1 2 (c 1 = 1) (c 1 =.1) 3.1 3.3 1 3 3.8 2.98 3.4.5 2.3 1 2 (c 1 =.444) (c 1 = 1) (c 1 =.1) 2.6 1 4 Table 4: Example 2. Numerical data of Figure 3 (case 1: m = 1, c 1 > ). Case 2 Varing c 2 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 2.97 3.1.1 4.3 1 2 (c 2 = 1) (c 2 =.1) 3.7 2.4 1 3 3.12 2.99 3.5.5 2 1 2 (c 2 =.6466) (c 2 = 1) (c 2 =.1) 4 1 5 2.997 3.1.1 4.3 1 3 (c 2 = 1) (c 2 =.1) 3.3 1 6 3.11 2.9985 3.5.5 2.15 1 3 (c 2 =.65114) (c 2 = 1) (c 2 =.1) 3.6 1 6 Table 5: Example 2. Numerical data of Figure 3 (case 2: m = 2, c 1 = ). h Case 1 - c 1 =.5 Case 2 - Radau I Blow-up Rel. error Blow-up Rel. error.1 2.941795 2 1 2 2.999559 1.5 1 4.5 2.97341 1 2 2.999778 7.4 1 5.1 2.993979 2 1 3 2.999955 1.5 1 5.5 2.996983 1 3 2.999978 7.3 1 6 Table 6: Example 2. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with c 1 =.5, and case 2 with Radau I collocation points, c 1 =, c 2 = 2/3, for different initial stepsizes h. Case 1 Varing c 1 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 5.2 6.55.1 2.3 1 1 (c 1 = 1) (c 1 =.1) 5.7625 3.8 1 3 5.855 5.45 6.2.5 1.3 1 1 (c 1 =.3961) (c 1 = 1) (c 1 =.1) 3.6 1 3 5.71 5.89.1 3.1 1 2 (c 1 = 1) (c 1 =.1) 5.77 2.6 1 3 5.7856 5.735 5.84.5 1.8 1 2 (c 1 =.441) (c 1 = 1) (c 1 =.1) 1.4 1 4 Table 7: Example 3. Numerical data of Figure 4 (case 1: m = 1, c 1 > ). The blow-up time of the corresponding (exact) solution is assumed to be ˆt 5.78482. 2

Case 2 Varing c 2 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 5.67 6.1.1 7.4 1 2 (c 2 = 1) (c 2 =.1) 5.793 1.4 1 3 5.7834 5.73 5.95.5 3.8 1 2 (c 2 =.5286) (c 2 = 1) (c 2 =.1) 3 1 4 5.775 5.81.1 6 1 3 (c 2 = 1) (c 2 =.1) 5.785 3.1 1 5 5.7848 5.779 5.8.5 3.6 1 3 (c 2 =.514) (c 2 = 1) (c 2 =.1) 3.5 1 6 Table 8: Example 3. Numerical data of Figure 4 (case 2: m = 2, c 1 = ). The blow-up time of the corresponding (exact) solution is assumed to be ˆt 5.78482. h Case 1 - c 1 =.5 Case 2 - Radau I Case 2 - c 2 =.5 Blow-up Rel. error Blow-up Rel. error Blow-up Rel. error.1 5.694865 1.6 1 2 5.751797 5.7 1 3 5.788215 5.9 1 4.5 5.739117 7.9 1 3 5.767963 2.9 1 3 5.786612 3.1 1 4.1 5.775121 1.7 1 3 5.78137 6.5 1 4 5.784995 3 1 5.5 5.78132 8.1 1 4 5.78366 3 1 4 5.784875 9.5 1 6 Table 9: Example 3. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with c 1 =.5, and case 2 with Radau I collocation points, c 1 =, c 2 = 2/3, and c 1 =, c 2 =.5, for different initial stepsizes h. The blow-up time of the corresponding (exact) solution is assumed to be ˆt 5.78482. Case 1 Varing c 1 Extrapolation Intersection h min. t max. t Rel. error t Rel. error t Rel. error 1.593 1.722.1 7.8 1 2 (c 1 = 1) (c 1 =.5) 1.6436 1.4 1 3 1.6467 1.613 1.691.5 4.7 1 2 (c 1 =.444) (c 1 = 1) (c 1 =.5) 5.2 1 4 1.6367 1.6571.1 1.2 1 2 (c 1 = 1) (c 1 =.5) 1.6456 1.2 1 4 1.64595 1.646 1.6521.5 7 1 3 (c 1 =.461) (c 1 = 1) (c 1 =.5) 6.6 1 5 Table 1: Example 4. Numerical data of Figure 5 (case 1: m = 1, c 1 > ). The blow-up time of the corresponding (exact) solution is assumed to be ˆt 1.645842. 21

Case 2 Varing c 2 Extrapolation h min. t max. t Rel. error t Rel. error.1 1.6445 (c 2 =.95) 1.649 (c 2 =.33) 2.7 1 3.5 1.6456 1.6476 (c 2 =.9) (c 2 =.3) 1.2 1 3 1.64648 3.9 1 4.1 1.64577 1.64622 (c 2 =.9) (c 2 =.3) 2.7 1 4.5 1.64581 1.6467 (c 2 =.85) (c 2 =.28) 1.6 1 4 1.64586 1.4 1 5 Table 11: Example 4. Numerical data of Figure 5 (case 1: m = 2, c 1 = ). The intersection method can not be applied because there are more than one intersection. The blow-up time of the corresponding (exact) solution is assumed to be ˆt 1.645842. h Case 1 - c 1 =.5 Case 2 - Radau I Blow-up Rel. error Blow-up Rel. error.1 1.6387 4.3 1 3 1.646991 7 1 4.5 1.642843 1.8 1 3 1.64654 4.2 1 4.1 1.645172 4.1 1 4 1.645985 8.7 1 5.5 1.645491 2.1 1 4 1.645919 4.7 1 5 Table 12: Example 4. Numerical estimations and relative errors of the blow-up time of collocation solutions in case 1 with c 1 =.5, and case 2 with Radau I collocation points, c 1 =, c 2 = 2/3, for different initial stepsizes h. The blow-up time of the corresponding (exact) solution is assumed to be ˆt 1.645842. 22