Some exact results for nonlinear diffusion with absorption

Simple exact solutions and first integrals are obtained fornonlinear diffusion incorporating absorption. These are obtainedby the standard techniques of separation of variables and theuse of invariant one-parameter group transforma-tions to reducethe governing partial differential equation to various ordinarydifferential equations. For two of the equations so obtained,first integrals are deduced which subsequently give rise toa number of explicit simple solutions. As with all special solutionsof nonlinear partial differential equations, the associatedinitial and boundary conditions are imposed by the particularfunctional form of the solution and irrespective of their physicalapplicability, simple exact solutions are always important,because one of the key features of nonlinearity is the rangeand variety of response which is often bizarre and unexpected,but which is frequently embodied in the simplest of exact solutions.Many of the solutions obtained here are illustrated graphicallywith particular reference to the phenomena of ‘extinction’and ‘blow-up’ and in general demonstrate a widevariety of differing physical response embodied in the disposableconstants, which is characteristic of nonlinear theories.