Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $, between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $.

In this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameter ɛ ∊ (0,1]. In the nonrelativistic limit regime, the small ɛ produces high oscillations in exact solutions with wavelength of (ɛ2) in time. The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time, with both the nonlinear and linear subproblems exactly integrable in time and, respectively, Fourier frequency spaces. The method is fully explicit and time reversible. Moreover, we establish rigorously the optimal error bounds of a second-order TSFP for fixed ɛ = (1), thanks to an observation that the scheme coincides with a type of trigonometric integrator. As the second task, numerical studies are carried out, with special efforts made to applying the TSFP in the nonrelativistic limit regime, which are geared towards understanding its temporal resolution capacity and meshing strategy for (ɛ2)-oscillatory solutions when 0 < ɛ « 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic (ɛ−2Δt2) temporal discretization error bound (Δt refers to time step). On the other hand, the temporal error bounds for most trigonometric integrators, such as the well-established Gautschi-type integrator in, are (ɛ−4Δt2). Thus, our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous or numerical, are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.

In this paper, stabilized Crank-Nicolson/Adams-Bashforth schemes are presented for the Allen-Cahn and Cahn-Hilliard equations. It is shown that the proposed time discretization schemes are either unconditionally energy stable, or conditionally energy stable under some reasonable stability conditions. Optimal error estimates for the semi-discrete schemes and fully-discrete schemes will be derived. Numerical experiments are carried out to demonstrate the theoretical results.

In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

We investigate the multidimensional non-isentropic Euler–Poisson (or full hydrodynamic) model for semiconductors, which contain an energy-conserved equation with non-zero thermal conductivity coefficient. We first discuss existence and uniqueness of the non-constant stationary solutions to the corresponding drift–diffusion equations. Then we establish the global existence of smooth solutions to the Cauchy problem with initial data, which are close to the stationary solutions. We find that these smooth solutions tend to the stationary solutions exponentially fast as $t\rightarrow+\infty$.

In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

Recent papers have shown that ${{C}^{1}}$ maps $F:\,{{\mathbb{R}}^{2}}\,\to {{\mathbb{R}}^{2}}$ whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or $F$ is a polynomial. Specifically, $F\,=\,(u,\,v)$ must take the form

$$u\,=\,ax\,+\,by\,+\,\beta \phi (\alpha x\,+\,\beta y)\,+\,e$$

$$v\,=\,cx\,+\,dy\,-\,\alpha \phi \,(\alpha x\,+\,\beta y)\,+\,f$$

for some constants $a,\,b,\,c,\,d,\,e,\,f,\,\alpha ,\,\beta $ and a ${{C}^{1}}$ function $\phi $ in one variable. If, in addition, the function $\phi $ is not affine, then

1

$$\alpha \beta (d\,-\,a)\,+\,b{{\alpha }^{2}}\,-\,c{{\beta }^{2}}\,=\,0.$$

This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.

We will show the existence, uniqueness and regularity of global solutions for the Cauchy problem for nonlinear evolution equations with the damping term .

As an application of our results, we give the global solvability and regularity of the mixed problem with Dirichiet boundary conditions:

In this paper we use a particular kind of weighted Sobolev space and pseudo-differential operators to study H3s propagation of singularities for the solution u ∊ Hs of the equations with second order.