الملخص الإنجليزي
du du 24u The evolution equation ay+B + y2 = f(u;m,k)u for the unknown u(x,t), dtdx 'ex2 depending on the space coordinate, x and the time t is studied. Here a,b,y,m,k are constants. Different forms of this equation are relevant to many applications which range from thermal convection to water waves to sand dunes to blood circulation in the human body. When B=y=0, the equation is studied extensively. The solution is obtained subject to an initial condition. Two particular examples are studied in detail. The first one is du/dt = m+klul u and the second is dul dt =[m+k(u-a)(u-1)]u in which a is a real constant. In each case, the dependence of the solution on the parameters and the form of the function f(u;m,k) is clarified. It is found that for real values of ,p, m, k the equation has solutions which can decay with time or grow to a finite asymptotic limit. The bifurcation curve for each case was used to identify the equilibrium solutions and study the stability of every solution. The information found in the study of the equilibrium solutions and their stability is used to show that the limiting values of the solutions correspond to the stable solution in each case. For certain values of the parameters, the solutions are unstable and these cases are found to correspond to the situations when the solution suffers a burst i.e. the solution increases to infinity for a finite value of the time, When, m, k are complex, it is found that the nature of the solution can change with the change in the imaginary parts of the parameters m,k. When y=0, a solution that is oscillatory in x is examined to deduce some results about the presence of the second spatial derivative on the solution. It is deduced that the influence of diffusion has the effect of stabilizing such a solution.
