We show how the more delicate properties of Sobolev spaces naturally arise in problems involving linear and nonlinear elliptic equations on smooth domains. We also show they arise in the theory of domain variation.

It is well known that nonnegative solutions of the heat equation defined in a strip $Q=\mathbf{R}\times (0,T)$ are uniquely determined by their initial data. In this paper we construct a nonnegative solution $H(x,t)$ of the nonlinear heat equation $$ u_t=(u^m)_{xx}- u^p, \quad 1<p<m, \tag {E} $$ which takes on zero initial data, $H(x,0)\equiv 0$, and is nontrivial for $t\ge \tau>0$. This special solution has the following properties: (i) $H(x,t)$ is bounded for $t\ge \tau>0$. (ii) There exists a constant $r>0$ such that $$ \begin{align} & H(x,t)>0 \quad \rm {if }\quad rt^{-\beta} <x<\infty,\\ & H(x,t)=0 \quad \rm{if }\quad -\infty <x<rt^{-\beta},\end{align} $$ where $\beta=(m-p)/2(p-1)$. (iii) It has the self-similar form $H(x,t)= t^{-\alpha}f(xt^{\beta}),$ where $0\le f(\eta) \le c$ is supported in the interval $r\le \eta<\infty$. A similar phenomenon holds in several space dimensions with radial symmetry, where it describes a form of focussing. On the other hand, such a nonuniqueness does not occur for other ranges of $m$ and $p$. In particular, it does not happen for linear diffusion, $m=1$.

A one-dimensional Sobolev-type inequality supplemented by a Prüfer transformation argument is used to derive upper and lower bounds for the eigenvalues of regular, self-adjoint second-order eigenvalue problems. These inequalities are shown to have applications to counting eigenvalues in the intervals $\scriptstyle (-\infty,\lambda]$, estimating eigenvalue gaps, Liapunov inequalities, and de La Valée Poussin-type inequalities.

We study the properties of the solutions of scalar conservation laws with a source term, assuming that the flux is convex and that the initial value has compact support. We show that their asymptotic profile consists in a number of rarefaction waves divided by regions where the solutions oscillate around an unstable zero of the source term.

We consider the initial value problems of the FitzHugh-Nagumo Equation (FHN) $$ \begin{align} \frac{ \partial v}{\partial t} & = Dv_{xx}-(v-a)v(v-1)-w+(I_i+\epsilon t)+g_1(I_i+ \epsilon t,t)\epsilon ^{\alpha +1} \quad \text{ for } 0<x <1,\ t>0 \\ \frac{ \partial w}{\partial t} & = bv-b\gamma w +g_2(I_i+\epsilon t,t)\epsilon ^{\alpha +1} \quad \text{ for } 0<x <1,\ t>0 \\ \frac{\partial v}{ \partial x} & = 0,\ \ \ \frac{\partial w}{ \partial x}=0 \quad \text{ for }x=0,1,\ t>0 \tag 0.1 \end{align} $$ where $\alpha \geq 0,$ $g_1$ and $g_2$ are analytic in both variables and periodic in $t$ with the period ${{2\pi }\over{\omega }}.$ We are interested in the delayed bifurcation phenomena of the system where the solutions have the property that while the linear stability of the solutions changes from stable to unstable after a critical point, the solutions remain dynamically stable (stay close to unstable solutions) until a point well above the critical point. In this work, we consider the delayed bifurcations in the FitzHugh-Nagumo Equation with the presence of some periodic forcing which models the membrane activities of the giant axon of a squid under certain external periodic influences. We show that if the frequency $\omega $ of $g_1,g_2$ and the Hopf bifurcation frequency of the FHN system are not in the ratio of 2:$n,$ then the delayed bifurcations occur.

In this paper we prove some results, which give the boundedness of minimizers of some degenerate functionals of Calculus of Variations. We consider also a "limit case".

Most of the existing results on global stability of delayed nonautonomous Lotka-Volterra type systems require that such systems have saturated equilibria, which is rather unnatural. In this paper, using a Lyapunov function, we obtain global stability results for such systems even if they have no saturated equilibria.

In this paper, the existence of solutions to a class of retarded functional differential equations is established along with a result concerning the continuous dependence of solutions on a parameter. These results generalize theorems concerning existence and continuous dependence in a Lebesgue integral setting to a Henstock-Kurzweil integral setting.

We use the methods of [24, 15] to provide an extension of results presented in [7, 16, 24] to derive existence of positive solutions to a general system of periodic parabolic equations which arise in the study of population dynamics. We will consider the following system with mixed boundary conditions. $$ \begin{align} u_t(x,t)-d_1(x,t)\Delta u(x,t)&= u(x,t)M(x,t,u,v) \\ v_t(x,t)-d_2(x,t)\Delta v(x,t)&= v(x,t)N(x,t,u,v). \end{align} $$

A classic solution to the Cauchy problem for the damped Boussinesq equation $u_{tt}-2Bu_{txx}=-\alpha u_{xxxx}+u_{xx}-\beta(u^2)_{xx}$, $x\in\Bbb R^1$, $t>0$, $\alpha, B=\text{const}>0$, $\beta=\text{const}\in\Bbb R^1$, with small initial data is constructed by means of the application of both the spectral and perturbation theories. Large time asymptotics of this solution are obtained. Its main term accounts for two solitons traveling in opposite directions. Each of them is governed by the Burgers equation with a transfer.