The Lie-group formalism is applied to investigate the symmetries of the modified Boussinesq system with variable coefficients. We derived the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The reduced systems of ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.

In this paper, we study the existence of nontrivial solutions for the problem -Δu=f(x,u,v)+h_1(x) in Ω, -Δv=g(x,u,v)+h_2(x) in Ω, u=v=0 on ∂Ω, where Ω is bounded domain in R^N and h_1,h_2∈L^2(Ω). The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g: \lim_{s,|t|→+∞}\frac{f(x,x,t)}{s}=\lim_{|s|,t→+∞}\frac{g(x,s,t)}{t}=λ_+, uniformly on Ω, \lim_{-s,|t|→+∞}\frac{f(x,x,t)}{s}=\lim_{|s|,-t→+∞}\frac{g(x,s,t)}{t}=λ_-, uniformly on Ω, where λ_+, λ_-∉{0}∪ σ(-Δ), σ(-Δ) denote the spectrum of -Δ. The cases (i) where λ_+=λ_ and (ii) where λ_+≠λ_- such that the closed interval with endpoints λ_+, λ_- contains at most one simple eigenvalue of -Δ are considered.

The existence and uniqueness of the solutions for the Boltzmann equations with measures as initial value are still an open problem which is posed by P. L. Lions (2000). The aim of this paper is to discuss the Cauchy problem of the system of discrete Boltzmann equations of the form ∂_tf_i+(f^{m_i}_i)_x=Q_i(f_1,f_2,...,f_n), (m_i > 1, i=1,...,n) with non-negative finite Radon measures as initial conditions. In particular, the existence and uniqueness of BV solutions for the above problem are obtained.

In this paper we prove the regularity, exponential stability of global solutions and existence of uniform compact attractors of semiprocesses, generated by the global solutions, of a two-parameter family of operators for a nonlinear one-dimensional non-autonomous equation of viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.