DEVELOPMENT OF SINGULARITIES IN SOLUTIONS OF A HYPERBOLIC SYSTEM

We consider a special type of a hyperbolic system and show that classical solutions blow up in finite time even for small initial data. 2000 Mathematics Subject Classification. 35L45.

1. Introduction.For the system of nonlinear elasticity u t (x, t) = ϕ v(x, t) v x (x, t), v t (x, t) = u x (x, t), (1.1) it is well known that C 1 -solutions break down in finite time however smooth and small the initial data are.This was shown by Lax [4] in 1964.In his work, the author studied (1.1), for ϕ > 0 and ϕ > 0, and established a blowup result.MacCamy and Mizel [7] in 1967 considered the same system and proved a similar result, allowing ϕ to change sign.They also showed, under appropriate conditions on ϕ, that there are x-intervals, for which the solution must exist for all time even though it blows up for values of x outside these intervals.
Messaoudi [9] discussed the following system: which models a transverse motion of a string with variable density.He showed that C 1 -solutions develop singularities in finite time if the initial data are taken with large enough gradients.He also discussed, in [8], a system with dissipation of the form which describes heat propagation in materials that predict finite propagation speed.This phenomenon is called second sound.Here θ is the difference temperature and q is the heat flux.He studied the Cauchy problem and proved a blowup result of the classical solutions.We should note that, for λ constant and c(θ) = −1, (1.3) reduces to a system describing steady shearing flows in nonlinear viscoelastic fluids.This problem was studied by Slemrod [11] and a blowup result for classical solutions has been established.A similar problem was also discussed by Nishibata [10], Kosiński [3], and Zheng [12] and results concerning global existence and nonexistence have been accomplished.
For more general systems, it is worth mentioning the work of Li et al. [6], in which they discussed u t (x, t) = A u(x, t) u x (x, t), (1.4) associated with decaying initial data.Here u : I × (0,T ) → R n is a vector-valued function, A is an (n×n)-matrix, and I is an interval (bounded or unbounded).They proved a global C 1 -solution for the Cauchy problem if, in addition to the local strict hyperbolicity condition, (1.4) is weakly linearly degenerate and the initial data satisfy, for µ > 0, |} is small enough.They also established a blowup result to C 1 -solutions for nonweakly linearly degenerate systems.As they pointed out, their work generalizes their result of [5] to the case of initial data with no compact support but they possess certain decay properties.
In this work, we are concerned with a quasilinear hyperbolic system of the form where the constant a ≠ 0. In addition to its importance from the mathematical technique point of view, this system can be regarded as a relative generalization of the one-dimensional wave equation in the sense if a = 0, (1.5) reduces to (1.1).We will consider (1.5) together with initial conditions and show that C 1 -solutions blowup even for small initial data.Our result cannot be directly deduced from the results of [6] since we do not impose the same conditions regarding the size and the regularity of the initial data (cf.[6, Theorem 1.2] and Theorem 3.1 below).This work is divided into two parts.In part one we state, without proof, a local existence theorem.In part two our main result is stated and proved.

Local existence.
We consider the following Cauchy problem where a ≠ 0 and ϕ is a function satisfying Proposition 2.1.Assume that ϕ is a C 1 function satisfying (2.4) and let u 0 and v 0 in H 2 (R) be given such that (2.5) Then the problem (2.1), (2.2), and (2.3) has a unique local solution (u, v), on a maximal time interval [0,T ), satisfying This result can be proved by applying a classical energy argument [1] or the nonlinear semigroup theory [2].
Remark 2.2.The functions u, v are C 1 functions by the standard Sobolev embedding theory.
3. Formation of singularities.We introduce the quantities and the differential operators where The following lemma shows, for initial data appropriately chosen, that r , s, and ρ are well defined and |v(x, t)/(1 + au(x, t))| is uniformly bounded.
Theorem 3.1.Let a and ϕ be as in Proposition 2.1.Then there exist initial data in H 2 (R) satisfying (2.5), for which av(x, t)

) and |v(x, t)/(1 + au(x, t))| is uniformly bounded on R × [0,T ).
Proof.We first choose δ > 0 such that if then Of course, this is possible by taking δ small enough.Then the continuity of u, v, and ϕ implies that there exists T ≤ T , such that (3.3) holds on R × [0,T ).Let T 0 := sup{T : (3.3) holds for all x ∈ R, t ∈ [0,T )}.We have two cases, either T 0 = T , this completes the proof.Or T 0 < T ; in this case we estimate We recall that, unless otherwise stated, α, β, ρ, and ϕ are functions of v/(1 + au).
Proof.We take an x-partial derivative of (3.7) to get which, in turn, implies