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We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation $\partial_{t}^2u = u\partial_x(u \partial_x u)$, which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude. [ABSTRACT FROM AUTHOR]

We study the final value problem for the nonlinear Schrödinger equations in one or two space dimensions with the gauge-invariant nonlinearity of the critical long-range power and nonresonant polynomial nonlinearities of the same power, which are not gauge invariant. We construct a unique global solution (for positive time) which is asymptotic for large time to a given profile with sufficiently small norm in lower order Sobolev spaces with decay at the spatial infinity and at the zero frequency. [ABSTRACT FROM AUTHOR]

The well posedness of the Cauchy problem for the operator P= D t2– D x a( t, x) n D x, $t,x \in \mathbb{R}$ with data on t=0 is studied assuming a ∈ C N( $[0,T];\gamma^{(s_0)}$( R)), s0>1 and sufficiently close to 1, a( t, x)≥ 0. Well posedness is proved in Gevrey classes γ( s), for $1\leq s< \frac{Nn^2}{2(N+2n)}$, n≥ n0. Keywords: Partial differential equations, Cauchy problem, Well posedness [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for second order hyperbolic equations with non negative characteristic form of two independent variables. We show that for such equations in divergence-free form, the Cauchy problem is well posed in the Gevrey class of order less than 5/2. [ABSTRACT FROM AUTHOR]