Existence of similarity profiles for diffusion equations and systems

We study the existence of similarity profiles for diffusion equations and reaction diffusion systems on the real line, where the different nontrivial limits are imposed for \( x \rightarrow -\infty \) $x\to -\infty$ and \(x \rightarrow +\infty \) $x\to +\infty$ . These similarity profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory.