We mean here the p-Laplacian equation (PLE for short) studied, for instance, in [20] and [23], which is the following nonlinear degenerate parabolic evolution equation (2.1) ∂u∂t=div|∇u|p−2∇u,p>2,t>0, 2,\hspace{1em}t>0,\]]]> subject to the initial condition (2.2) u(x,0)=δ(x), where u:=u(x,t) , with x∈Rd,d≥1 , is a scalar function defined on Rd×R+ . The Cauchy problem (2.1)–(2.2) admits a unique nonnegative fundamental solution: it is a function u≥0 solving (2.1)–(2.5) in a weak sense (see [20] and [23] for the detailed definition of weak solution to PLE). This solution is given by the following probability density function (2.3) u(x,t)=t−kc−q‖x‖tk/dpp−1+p−1p−2, where k:=p−2+pd−1,q:=p−2pkd1p−1 and c:=c(p,d) is a constant determined by the condition ∫u(x,t)dx=1 . By setting β=pp−1,γ=p−1p−2,α=kd,C=cp−1p−2 and c=(c/q)p−1p , the Barenblatt-type solution (1.2) coincides with (2.3).

The Nonlocal Porous Medium Equation (NPME), studied in [3] and [4], is the following degenerate nonlinear and nonlocal evolution equation (2.4) ∂u∂t=div|u|∇ν−1(|u|m−2u),m>1,ν∈(0,2],t>0, 1,\hspace{0.1667em}\nu \in (0,2],\hspace{0.1667em}t>0,\]]]> subject to the initial condition (2.5) u(x,0)=u0(x). The pseudo-differential operator ∇ν−1 is the fractional gradient denoting the nonlocal operator defined as ∇ν−1u:=F−1(iξ‖ξ‖ν−2Fu) . This notation highlights that ∇ν−1 is a pseudo-differential (vector-valued) operator of order ν−1 . Equivalently, we can define ∇ν−1 as ∇(−Δ)ν2−1 , where (−Δ)ν2u=F−1(‖ξ‖νFu) is the fractional Laplace operator, i.e., a Fourier multiplier with the symbol ‖ξ‖ν . For ν=2 , (2.4) becomes the classical nonlinear porous medium equation (2.6) ∂u∂t=div|u|∇(|u|m−2u)=div(m−1)|u|m−1∇u. If we restrict our attention to nonnegative solution u(x,t) , equation (2.6) becomes (2.7) ∂u∂t=m−1mΔ(um), which is usually adopted to model the flow of a gas through a porous medium.

Let ν∈(0,2] and m>1 1$]]>. A weak solution, in the sense of Definition 1 in [4], is given by (2.8) u(x,t)=Ct−dα1−k2ν‖x‖2t2α+ν2(m−1), where α:=1d(m−1)+ν , k:=dΓ(d/2)(d(m−1)+ν)2νΓ(1+ν2)Γ(d+ν2) and C:=Γ(d2+ν2(m−1)+1)kdνπd2Γ(ν2(m−1)+1). Furthermore, u(x,t) is the pointwise solution of equation (2.4) for ‖x‖≠k−1νtα . The link between (2.8) and random flights has been investigated in [8]. For ν=2 , the solution (2.3) becomes the Barenblatt–Kompanets–Zel’dovich–Pattle solution of the porous medium equation (2.7) supplemented with the initial condition u(x,0)=δ(x) (see, for instance, [33]).

The function (1.2) reduces to (2.8) for γ=ν2(m−1),β=2 and c=1/k2ν .

It is well known that the fundamental solution of the Euler–Poisson–Darboux (EPD) equation (2.9) ∂2u∂t2+d+2ν−1t∂u∂t=c2Δu,ν>0,t>0,c>0, 0,\hspace{2.5pt}t>0,\hspace{2.5pt}c>0,\]]]> has the form (2.10) u(x,t)=Γ(ν+d2)πd/2Γ(ν)1(ct)d1−‖x‖2(ct)2+ν−1 and therefore it belongs to the family of probability density functions with compact support (1.2) with β=2 , α=1 , k=d and γ=ν−1 . There is a wide literature about the EPD equation and its applications. We refer to Bresters [5] for the construction of weak solutions of the initial value problem for the EPD equation based on distributional methods. We recall that, in the one-dimensional case, the solution to the Cauchy problem (2.11) ∂2u∂t2+2ξt∂u∂t=c2∂2u∂x2,u(x,0)=f(x),∂u∂t(x,t)|t=0=0, can be represented as the Erdélyi–Kober fractional integral (see definition (3.4) below) of the D’Alembert solution of the wave equation (see [10]) (2.12) ∂2w∂t2=c2∂2w∂x2,w(x,0)=f(x),∂w∂t(x,t)|t=0=0.

This means that (2.13) u(x,t)=2Beta(ξ,12)∫01(1−y2)ξ−1f(x+yct)+f(x−yct)2dy. The first probabilistic interpretation of this analytic representation, discussed by Rosencrans [31] and more recently by Garra and Orsingher [12], is the following one: solution (2.13) can be written as (2.14) u(x,t)=Ef(x+U(t))+f(x−U(t))2, where (2.15) U(t)=U(0)∫0t(−1)N(s)ds, and (N(t))t≥0 is the nonhomogeneous Poisson process with rate λ(t)=ξt , U(0) is a uniformly distributed r.v. on {−c,c} (furthermore (N(t))t≥0 and U(0) are supposed independent). By means of the general Proposition 1 (see the next section), we here obtain a new interesting probabilistic interpretation of the fundamental solution of the EPD equation.

Moreover, we have the following interesting picture that underlines the role of the Barenblatt-type solution as a bridge between nonlinear and linear PDEs: •

The Erdélyi–Kober fractional integral of the solution of the D’Alembert equation leads to the solution of the Euler–Poisson–Darboux equation under the same initial conditions.

There is a wide literature about the probabilistic interpretation of linear space and time-fractional diffusive equations (see, e.g., [24, 1, 6, 27] and the references therein). On the other hand, a probabilistic approach to time-fractional nonlinear diffusive-type equations is still completly missing. Recently, the existence and uniqueness of compactly supported solutions for time-fractional porous medium equations has been investigated (see, e.g., [30]). However, up to our knowledge, it is not possible to find an explicit form of the Barenblatt-type solution. On the other hand, time-fractional diffusive equations are actracting an increasing interest in the literature. Explicit Barenblatt-type solutions for nonlinear time-fractional equations can play a relevant role for future studies in this context. We here consider a new family of nonlinear time-fractional diffusive equations admitting a Barenblatt-type solution of the form (1.2).

Let us consider the following nonlinear diffusive equation (2.16) 1t2ν∂νu∂tν+1tν∂2u∂x2+∂u∂x2=0, where ∂ν∂tν denotes the Riemann–Liouville derivative ∂νf(x,t)∂tν=1Γ(1−ν)∂∂t∫0t(t−s)−νf(x,s)ds,ν∈(0,1), and f(x,·) is a suitable well-behaved function (see [22] for details about the functional setting).

We start our analysis from a simple ansatz: equation (2.16) admits a solution in the form (2.17) u(x,t)=C1tν−C2x2t3ν,ν∈0,1/3∖{1/4},(x,t)∈R×R+, where C1 and C2 are real constants that we are going to find. We now directly check the correctness of this conjecture. We first recall that, for ν>0 0$]]> and β>−1 -1$]]>, (2.18) ∂ν∂tνtβ=Γ(1+β)Γ(1+β−ν)tβ−ν. Then, by substituting (2.17) in (2.16) and by applying (2.18), we have (2.19) Γ(1−ν)Γ(1−2ν)C1t4ν−Γ(1−3ν)Γ(1−4ν)C2x2t6ν−2C2t4ν+4C22x2t6ν=0 and balancing similar terms we have that (2.20) Γ(1−ν)Γ(1−2ν)C1−2C2=0,4C22−Γ(1−3ν)Γ(1−4ν)C2=0. Therefore (2.21) C1=12Γ(1−3ν)Γ(1−4ν)Γ(1−2ν)Γ(1−ν),C2=14Γ(1−3ν)Γ(1−4ν).

Observe that the constraint on the real order of derivation ν∈0,13 in (2.17) is due to the application of (2.18).

Moreover, ν≠1/4 because of the coefficients appearing in the solution (depending on the Euler Gamma functions). We can conclude that equation (2.16) admits a solution of the form (2.17) if ν∈(0,1/3)∖{1/4} . Therefore, in particular, under the same constraints on ν, we can say that the time-fractional equation (2.16) admits a solution of the form (2.22) u(x,t)=C1tν1−C2C1x2t2ν+, where C1 and C2 are given by (2.21).

Obviously, the solution (2.22) is not normalized but it is a Barenblatt-type solution belonging to the general family considered in this paper. A systematic study of equation (2.16) should be an object of further investigation, both from the physical and mathematical points of view. We conjecture that this is a source-type solution of the nonlinear time-fractional equation (2.16); nevertheless a full rigorous analysis should be developed, but this is beyond the aims of this paper. The fractional equation (2.16) can be viewed as a hybrid between a diffusive equation with singular time-dependent coefficients (in some way similar to the EPD equation) and a nonlinear time-fractional porous medium type equation.