2023/10/11 by Iagar, Razvan Gabriel, Laurençot, Philippe
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.07270
Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption∂t u=Δum-|x|σuq,posed for (t,x)∈(0,∞)×ℝN, with m>1, q∈(0,1) and σ=σc:=2(1-q)/(m-1) is proved. Looking for radially symmetric solutions of the formu(t,x)=e-αtf(|x|eβt), α=(2)/(m-1)β,we show that there exists a unique exponent β^*∈(0,∞) for which there exists a one-parameter family (uA)A>0 of solutions with compactly supported and non-increasing profiles (fA)A>0 satisfying fA(0)=A and fA'(0)=0. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when σ∈ (0,σc).