The initial value problem$$\eqalign{u\sb{t} &= \Delta u - u + H(u-a),\cr u(x,0) &= u\sb0(x),\cr}$$for $x\in \IR\sp2$ and $t > 0$ is studied in this work. We prove existence of traveling wave solutions by studying the behavior of the level curves $u = a$ as t increases. In Chapter 3 it is shown that traveling wa e solutions with radially symmetric initial values have velocity 0, using this fact we give these solutions using Bessel functions and then we prove that these solutions are unstable. Finally, in the last chapter we present some results which were obtained using numerical approximations
