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Register a new userNonnegative solutions for a long-wave unstable thin film equation with convection
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We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finally, we present numerical simulations of solutions.
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In this technical report, we consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finally, we present numerical simulations of solutions.
We investigate the large time behavior of compactly supported solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.
In this paper we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we prove also prove smoothness and exponential decay at infinity.
We study short--time existence, long--time existence, finite speed of propagation, and finite--time blow--up of nonnegative solutions for long-wave unstable thin film equations $h_t = -a_0(h^n h_{xxx})_x - a_1(h^m h_x)_x$ with $n>0$, $a_0 > 0$, and $a_1 >0$. The existence and finite speed of propagation results extend those of [Comm Pure Appl Math 51:625--661, 1998]. For $0<n<2$ we prove the existence of a nonnegative, compactly--supported, strong solution on the line that blows up in finite time. The construction requires that the initial data be nonnegative, compactly supported in $R^1$, be in $H^1(R^1)$, and have negative energy. The blow-up is proven for a large range of $(n,m)$ exponents and extends the results of [Indiana Univ Math J 49:1323--1366, 2000].
We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ and potential periodic in $t$ may have solutions with exponentially increasing as $ t to infty$ norm $H^1({mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({mathbb R}^3_x)$ norm of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence ${Y_k(ntau_k)}_{n = 0}^{infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < tau_k <1.$
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