A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions
Razvan Gabriel Iagar, Diana-Rodica Munteanu · Journal of Mathematical Analysis and Applications · 2024
This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption ∂ t u = Δ u m − | x | σ u p , posed for ( x , t ) ∈ R N × ( 0 , ∞ ) , N ≥ 1 , and in the range of exponents 1 0 . We give a complete classification of (singular) self-similar solutions of the form u ( x , t ) = t − α f ( | x | t − β ) , α = σ + 2 σ ( m − 1 ) + 2 ( p − 1 ) , β = p − m σ ( m − 1 ) + 2 ( p − 1 ) , showing that their form and behavior strongly depends on the critical exponent p F ( σ ) = m + σ + 2 N . For p ≥ p F ( σ ) , we prove that all self-similar solutions have a tail as | x | → ∞ of one of the forms u ( x , t ) ∼ C | x | − ( σ + 2 ) / ( p − m ) or u ( x , t ) ∼ ( 1 p − 1 ) 1 / ( p − 1 ) | x | − σ / ( p − 1 ) , while for m < p < p F ( σ ) we add to the previous the existence and uniqueness of a compactly supported very singular solution . These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper.