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We study the quasilinear non-local Benney System $$left{begin{array}{llll} iu_t+u_{xx}=|u|^2u+buv v_t+a(int_{mathbf{R}^+}v^2dx)v_x=-b(|u|^2)_x,quad (x,t)inmathbf{R}^+times [0,T],, T>0. end{array}right.$$ We establish the existence and uniqueness of strong local solutions to the corresponding Cauchy problem and show, under certain conditions, the blow-up of such solutions in finite time. Furthermore, we prove the existence of global weak solutions and exhibit bound-state solutions to this system.
Proving local well-posedness for quasilinear problems in pdes presents a number of difficulties, some of which are universal and others of which are more problem specific. While a common standard, going back to Hadamard, has existed for a long time,
We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ left{begin{array}{rlll} e^{-itheta}u_t&=&u_{xx}-|u|^2u-alpha g(v)u& v_t+(f(v))_x&=&alpha (g(v)|u|^2)_x& end{array}right. qqu
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the para
The paper studies the long time behavior of a system that describes the motion of a piece of elastic membrane driven by surface tension and inner air pressure. The system is a degenerate quasilinear hyperbolic one that involves the mean curvature, an
This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modu