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We study the initial value problem of the thermal-diffusive combustion systemu 1, t= u 1, x, x u 1 u 2 2, u 2, t= du 2, xx+ u 1 u 2 2, x R 1, for non-negative spatially decaying initial data of arbitrary size and for any positive constantd. We show that if the initial data decay to zero sufficiently fast at infinity, then the solution (u 1, u 2) converges to a self-similar solution of the reduced systemu 1, t= u 1, xx u 1 u 2 2, u 2, t= du 2, xx, in the large time limit. In particular, u 1 decays to zero like O (t 1/2 ), where> 0 is an anomalous exponent depending on the initial data, andu 2 decays to zero with normal rate O (t 1/2). The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self-similarity of the solutions in the large time limit.

Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.

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. Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\Phi=\Phi(u,v)$, equivalent to the $\L^1$ distance, which is almost decreasing i.e., $$ \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t&gt;s\geq 0,$$ for every pair of <i></i>-approximate solutions <i>u</i>, <i>v</i> with small total variation, generated by a wave front tracking algorithm. The small parameter <i></i> here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in <i>u</i> and in <i>v</i>. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the ${\vec L}^1$ norm. This provides a new proof of the existence of the