Evolution equations involving nonlinear truncated Laplacian operators
Matthieu Alfaro Isabeau Birindelli


Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities

    Holger F. Hofmann
    Submitted 6 July 2018 / Accepted 19 March 2019
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Matthieu Alfaro [IMAG, Univ. Montpellier, CNRS, Montpellier, France. E-mail: matthieu.alfaro@umontpellier.fr] and Isabeau Birindelli [Dipartimento di Matematica, Sapienza Università, Rome, Italy. E-mail:
isabeau@mat.uniroma1.it]. 

[2ex]











 We first study the so-called Heat equation with two families of elliptic operators which are fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equation with operators including the large” eigenvalues has strong similarities with  a Heat equation in lower dimension whereas, surprisingly, for operators including small” eigenvalues it shares some properties with some transport equations. In particular, for these operators, the Heat equation (which is nonlinear) not only does not have the property that disturbances propagate with infinite speed” but may lead to quenching in finite time. Last, based on our analysis of the Heat equations (for which we provide a large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.


Key Words: fully nonlinear elliptic operator, Heat equation, Cauchy problem, viscosity solutions, quenching phenomena, Fujita blow-up phenomena.


AMS Subject Classifications: 35K05 (Heat equation), 35K65 (Degenerate parabolic equations), 35L02 (First-order hyperbolic equations), 35C06 (Self similar solutions), 35D40 (Viscosity solutions).





§ INTRODUCTION



Let N≥ 2 be given. For u:^N→, say of the class C^2, we denote by

    λ_1(D^2u)≤⋯≤λ _N(D^2u)

the eigenvalues of the Hessian matrix D^2u. For 1≤ k<N we consider the fully nonlinear elliptic operators given by

    _k^- u:=∑_i=1^k λ_i(D^2u),

and

    _k^+ u:=∑_i=N-k+1^N λ_i(D^2u).

Notice that the case k=N leads to the linear situation _N^-u= _N^+u=Δ u, hence we will always suppose that k<N.

Our first main goal is to understand  the Heat equations

    ∂ _t u= _k ^-  u    in  (0,+∞)×^N,

and

    ∂ _t u=   _k^+ u   in  (0,+∞)×^N,

together with the associated Cauchy problems. As revealed below by our analysis, naming (<ref>) a Heat equation is controversial but, for the moment, we adopt this denomination.

Our second main goal is to analyze the  Fujita blow-up phenomena <cit.>, <cit.>, <cit.>, <cit.>, for the Cauchy problems associated with equations

    ∂ _t u= _k^- u+u^1+p   in  (0,+∞)×^N,

and

    ∂ _t u= _k ^+u+u^1+p   in  (0,+∞)×^N,

where p>0.



Let us mention that these highly degenerate elliptic operators have been introduced in the context of differential 
geometry, by Wu <cit.> and Sha <cit.>, in order to solve problems related to manifolds with partial 
positive curvature. In a related fashion, they appear in the analysis of mean curvature flow in arbitrary 
codimension performed by Ambrosio and Soner   <cit.>.

In the context of elliptic PDE they are already considered as an example of degenerate fully non linear operators 
in the User's guide <cit.>, but more recently both 
Harvey and Lawson in <cit.> and Caffarelli, Li and Nirenberg <cit.> have 
studied them in a completely new light.


Finally, in the very last years, some new results have been obtained on Dirichlet problems in bounded domains in 
relationship with the convexity of the domain, through the study of the maximum principle and the so called principal eigenvalue",  see <cit.> and  <cit.>. 


Notice also that, due to the links between the behavior of solutions to evolution equations (Fujita blow up phenomenon) and the existence of steady states (nonlinear Liouville theorems), see <cit.> e.g., the works of Birindelli, Galise and Leoni <cit.>  and  Galise <cit.> can be seen as a starting point for the present paper focused on evolution problems.

We  also wish to mention the very recent works of Blanc and Rossi that study
degenerate elliptic operators defined by 𝒫 _j u:=λ_j(D^2u) for some 1≤ j≤ N.  In other words, instead of considering the sum of the 
k smallest or k largest eigenvalues, they consider only the j-th eigenvalue of the Hessian matrix.
Even though they are different operators they share some analogies both in the definitions and in the difficulties 
that arise in studying them. These authors have considered both the steady state equation <cit.> and the evolution equation <cit.> in a bounded domain. They mainly focus on the well-posedness of such problems (in the viscosity sense), and their approximation by a two-player zero-sum game.

As far as we know, this work is the first analysis of evolution equations in ^N  involving the aforementioned nonlinear 
truncated Laplacian operators. Since we explore many directions, and collect results that we believe to be of 
equal importance, we take the liberty not to present a section with some so-called main results. Instead, we give 
below a rather detailed overview of the paper.

In Section <ref> we compute naive explicit solutions to the Heat equations (<ref>) and 
(<ref>). This can be seen as a warm-up, already revealing the importance of convexity/concavity of 
solutions.

In Section <ref> we inquire on the existence of self similar solutions to (<ref>) and 
(<ref>). A key observation is that when a solution u is one dimensional”, its Hessian has an eigenvalue of multiplicity (at least) N-1 and therefore, computing _k^± u reduces to localize the last eigenvalue, see assumption (<ref>). The outcomes are the following: for equation (<ref>) involving _k ^-, self similar solutions have algebraic decay as | x|→ +∞; for equation (<ref>) involving _k ^+ self similar solutions are the Heat kernels in lower dimension k<N which, in particular, have a L^1( ^N) norm which is  increasing in time like t^N-k/2.

In Section <ref>, we quote a result of Crandall and Lions <cit.> to obtain the global well-posedness, in the viscosity sense, of the Cauchy problems (<ref>) and (<ref>), and the local well-posedness, of the Cauchy problems (<ref>) and (<ref>), which we plan to study in the end of the paper.

In Section <ref> we inquire on radial solutions to the Cauchy problems (<ref>) and (<ref>). It turns out that the Heat equation (<ref>) involving _k ^- may not diffuse but transports. In other words, the operator _k^- shares some similarities with some first order operators. As a by product, we can construct a very surprising example of an initial data driving the solution to zero everywhere in finite time, see Example <ref>, which is referred as a quenching phenomena. On the other hand, and as already suspected since Section <ref>, the Heat equation (<ref>) in dimension N involving _k^+ behaves like the Heat equation in lower dimension k<N.

Finally, Section <ref> is devoted to the analysis of the Cauchy problems (<ref>) and (<ref>). We aim at determining the Fujita exponent p_F separating  sytematic blow-up when 0<p<p_F” from existence of global solutions when p>p_F” (see Section <ref> for a more precise statement). The proofs rely on the variety of special solutions to the Heat equations (<ref>) and (<ref>) collected in the previous sections. We prove that p_F=0 for (<ref>)  involving _k^-, whereas p_F=2/k for (<ref>) involving _k ^+. These facts were highly suspected from the previous sections but the proofs for (<ref>) are far from trivial: the proof of Theorem <ref> in particular requires the combination of the comparison principle, the subtle solutions of Example <ref> and a comparison between _k^+ u and Δ u which is available for radial and smooth solutions.


Let us mention that, from places to places, we have indicated some directions and presented some
preliminary computations that lead to partial conclusions or observations, and therefore raise some open problems. We have also tried to underline the variety of possible behaviors of the evolution equations under consideration by providing many examples of very different solutions.




§ EXPLICIT SOLUTIONS TO THE HEAT EQUATIONS




 §.§ Convex/concave functions of one variable


If, for some 1≤ i≤ N,

    u(t,x)=φ (x_i),    with φ a C^2 convex function,

then u solves (<ref>). Indeed D^2u=Diag  (0,0,...,φ”(x_i)_ith position,...,0,0). Since φ”(x_i)≥ 0 the N-1 smallest eigenvalues are 0, hence 𝒫_k^- u=λ_1(D^2u)+⋯+λ_k(D^2u)=0=∂ _t u.



Similarly if, for some 1≤ i≤ N,

    u(t,x)=φ (x_i),    with φ a C^2 concave function,

then u solves (<ref>).



 §.§ One variable travelling waves

For some 1≤ i≤ N, let

    u(t,x)=φ(x_i-ct),   with c≠0.

If φ is convex then, again, 𝒫 _k^- u=0 and thus we need -cφ'=0 and φ=cst already found above. On the other hand, if φ is concave then 𝒫_k^-u(t,x)=φ”(x_i-ct) and thus we need -cφ '=φ” that is
φ(z)=α-β e^-cz with β>0 to get the concavity. Hence we are equipped with 

    u(t,x)=α-β e^-c(x_i-ct),   α∈, β>0, c≠ 0,

solutions to (<ref>) that are planar travelling waves connecting α to -∞.



Similarly, we are equipped with 

    u(t,x)=α+β e^-c(x_i-ct),   α∈, β>0, c≠ 0,

solutions to (<ref>) that are planar travelling waves connecting α to +∞.



 §.§ Polynomial solutions
















For any A∈𝒮_N(), any x_0∈^N, any y∈^N, any C∈,

    u(t,x)=(∑_i=1^k λ_i(A))t+1/2 A(x-x_0)·(x-x_0)+(x-x_0)· y +C

solves (<ref>), whereas 

    u(t,x)=(∑_i=N-k+1^N λ_i(A))t+1/2 A(x-x_0)·(x-x_0)+(x-x_0)· y +C

solves (<ref>), since in the two above cases D^2u=A. Those solutions provide the sub and supersolutions used in the proof of <cit.>.



§ THE HEAT EQUATIONS: SELF SIMILAR SOLUTIONS


If u(t,x) solves (<ref>) or (<ref>) so does Cu(λ^2t,λ x), C>0, λ>0. We thus look after a nonnegative self similar solution in the form

    u(t,x):=1/t^βφ(| x|/√(t)),

for some φ=φ(r), β∈, and where | x| =(x_1^2+⋯+x_N^2)^1/2. We also require φ(0)=1, φ'(0)=0.

We immediately get 

    ∂ _t u=-β/t^β+1φ -1/2| x|/t^β+3/2φ '.

Next, after straightforward computations, we obtain the Hessian matrix

    D^2u=1/t^β+1/2(1/| x|φ'Id_N-(1/| x|φ' -1/t^1/2φ”)x/| x|⊗x/| x|).

Since x/| x|⊗x/| x| is a matrix of rank 1, 1/t^β+1/2| x|φ ' is an eigenvalue of D^2u with multiplicity (at least) N-1. By considering the traces of the matrices we see that the remaining eigenvalue has to be 1/t^β+1φ”. From now on, we assume

    φ”(r)≥1/rφ'(r),    for all  r>0,

which enables to compute 𝒫^± _k u. Notice that other assumptions than (<ref>) will be discussed in subsection <ref> and will reveal much less natural.



 §.§ Operator _k^-


Under assumption (<ref>), we have

    ^-_ku=∑ _i=1^kλ_i(D^2u)=k/t^β+1/2| x|φ',

and thus the Heat equation (<ref>) is transferred into  the  linear first order ODE  Cauchy problem

    φ'=-β2r/r^2+2kφ,  φ(0)=1,

which is solved as 

    φ(r)=(2k/r^2+2k)^β,

which in turn does satisfy (<ref>) if β≤ -1 or β≥ 0. In order to keep nonconstant and bounded solutions, we now restrict to β>0: going back to (<ref>), we are equipped, for any β>0, μ >0, with solutions

    u(t,x)=μ(1/| x| ^2+2kt)^β,

and also, for any β>0, μ>0, >0,

    u_(t,x)=μ(1/| x| ^2+2kt+)^β.


 Let 1≤ p≤ +∞. For any t>0, u(t,·) belongs to L^p(^N) as soon as β>N/2p and we have

    ‖ u(t,·)‖_L^p(^N)=μ C/t^β-N/2p,

with C=C(β,p,N,k)>0. In particular the L^∞ norm decreases like 1/t^β.





 §.§ Operator ^+_k


Under assumption (<ref>), we have

    ^+_ku=∑ _i=N-k+1^Nλ_i(D^2u)=1/t^β+1φ”+k-1/t^β+1/2| x|φ',

and thus the Heat equation (<ref>) is transferred into  the linear problem

    φ”+r^2+2(k-1)/2rφ'+βφ=0,   φ(0)=1,  φ'(0)=0.

One recognizes the ODE arising when looking after self-similar solutions to the Heat equation in dimension k<N. Hence, φ(r):=e^-r^2/4 solves the above problem provided that β=k/2, and does satisfy (<ref>). Hence, going back to (<ref>), we are equipped for any μ>0, with solutions

    u(t,x)=μ/t^k/2e^-| x| ^2/4t,    t>0,  x∈^N.

In particular notice that, for μ=(4π)^-N/2, we have ∫ _^Nu(t,x)dx=t^N-k/2→ +∞, as t→ +∞.

 For the self-containedness of the argument, we briefly discuss the problem (<ref>) when β≠k/2. Using a Sturm-Liouville approach, one can recast the ODE problem (<ref>) into an integral equation and prove the existence and uniqueness of a  local solution which moreover always satisfies φ”(0)=-β/k, and is global when β>0, see <cit.>.

If β=0, the solution is φ≡ 1.

If β<0, we claim that φ≥ 1: if not, from φ(0)=1, φ'(0)=0, φ”(0)>0, there must be a point r_0>0 where φ reaches a local maximum larger than 1; testing the equation at r_0 yields a contradiction. We get rid of these solutions which are larger than one.


Now, for β >0, β≠k/2,  writing φ(r)=e^-r^2/4ψ(r^2/4), we see that ψ has to solve 
 
    zψ”(z)+(k/2-z)ψ'(z)+(β-k/2)ψ(z)=0,   ψ(0)=1,  ψ'(0)=1-2β/k,

 and thus ψ(z)=_1F_1(k-2β/2,k/2,z), where _1 F_1(a,b,z) is 
the confluent hypergeometric function of first kind, or Kummer's function, see <cit.>. It is known that, when a is not a nonpositive integer, 

    _1F_1(a,b,z)∼Γ(b)/Γ(a)e^z/z^b-a,    as  z→ +∞.

This transfers, when 0<β<k/2, into φ(r)∼C/r^2β, for some C>0, as r→+∞,  and thus ∫ _0 ^+∞r^N-1φ(r)dr=+∞, so that these solutions are not admissible”. When β>k/2 and k-2β/2 is not a negative integer, the conclusion is again φ(r)∼C/r^2β, for some C>0 or C<0, as r→+∞, and these solutions are not admissible”. Last, when k-2β/2 is a negative integer, say -p, ψ(z) is the p-th generalized Laguerre polynomial, which is known <cit.> to change sign on (0,+∞), and thus these solutions are not admissible”.




 §.§ On assumption (<ref>)


The goal of this short subsection is to show that assumption (<ref>) is the one to be retained, as claimed above.

First, assuming the reverse inequality, namely

    φ”(r)≤1/rφ'(r),    for all  r>0,

we can still compute 𝒫^± _k u, where u(t,x) is given by the self-similar ansatz (<ref>). But, when dealing with operator _k^-, we now reach the second order ODE problem (<ref>), whose solution φ(r)=e^-r^2/4 does not satisfy (<ref>). Similarly, when dealing with operator _k^+, we now reach the first order ODE problem (<ref>), whose solutions φ(r)=(2k/r^2+2k)^β do not satisfy (<ref>). 

Next, we may only assume the existence of >0 such that

    φ”(r)≥1/rφ'(r),    for all  0<r<.

Then, dealing with _k ^-, we reach φ(r)=(2k/r^2+2k)^β, say for β>0, for which φ”(r)>1/rφ'(r) holds all along (0,+∞). In other words, we are back to assumption (<ref>). The same argument applies when dealing with _k ^+.
 
 Last, assuming (<ref>) only a small bounded interval (0,), we reach a contradiction as in the case  of assumption (<ref>).



§ WELL-POSEDNESS OF THE DIFFERENT CAUCHY PROBLEMS


For A∈𝒮_N(), we define F_k^-(A):=∑_i=1^k λ _i(A) and F_k^+(A):=∑_i=N-k+1^Nλ _i(A). From the min-max theorem for eigenvalues of real symmetric matrices, we have that, for any A,B∈𝒮 _N(),

    A≥ B ⟹λ_i(A)≥λ _i(B), ∀ 1≤ i≤ N ⟹ F_k^±(A)≥ F_k^±(B),

and that, for any A∈𝒮_N(), any c∈, 

    F_k^±(A+cId_N)-F_k^±(A)≥ kc.


This enables to quote <cit.>: for a initial data  u_0∈ UC(^N), the Cauchy problems associated with the Heat equations (<ref>) and (<ref>) admit a comparison principle and are globally well-posed, solutions being understood in the viscosity sense,  <cit.>, <cit.>, <cit.>, <cit.>. The proof follows the three main steps: first prove a comparison principle using a  dedoubling variable method, next construct polynomial sub and supersolutions in the spirit of subsection <ref>, last conclude by the Perron's method.

By a straightforward and classical modification of the above procedure, one can prove the well-posedness of the Cauchy problems associated with equations (<ref>) and (<ref>), at least locally in time. The main issue is then to determine if the local solution is global or blows up in finite time, which will be discussed in Section <ref>.



§ THE HEAT EQUATIONS: RADIAL SOLUTIONS OF THE CAUCHY PROBLEMS


We  consider the Cauchy problem (<ref>) or (<ref>) starting from a radial initial data u_0(x)=g(| x|), where g:[0,+∞)→. We suspect that u(t,·) remains radial for t>0 and therefore use the ansatz

    u(t,x)=ψ(t,| x|),

for some ψ=ψ(t,r). We compute the Hessian matrix and get

    D^2u=1/| x|(∂ _r ψ) Id_N-(1/| x|∂ _r ψ-∂_rrψ)x/| x|⊗x/| x|.

whose eigenvalues are 1/| x|∂ _r ψ with multiplicity (at least) N-1 and ∂_rrψ (see Section <ref>). From now on, guided by (<ref>) and subsection <ref>, we assume

    ∂ _rrψ(t,r)≥1/r∂ _r ψ(t,r),  for all  t>0, r>0,

which enables to compute 𝒫^± _k u.



 §.§ Operator ^-_k


Under assumption (<ref>), we have

    ^-_ku=∑ _i=1^kλ_i(D^2u)=k/| x|∂ _r ψ,

and thus the Heat equation (<ref>) is transferred into  the  linear transport equation

    ∂ _tψ=k/r∂ _r ψ,

that can be solved via the method of characteristics. Indeed, for r_0>0, we have

    d/dt[ψ(t,(r_0^2-2kt)^1/2)]=0,

and thus ψ(t,(r_0^2-2kt)^1/2)=ψ(0,r_0)=g(r_0) which is recast

    ψ(t,r)=g((2kt+r^2)^1/2).


Conversely, we need to check that assumption (<ref>) is satisfied. From (<ref>) we compute, assuming further regularity for g,

    ∂ _rrψ(t,r)-1/r∂ _r ψ(t,r)=r^2/2kt+r^2(g”(s)-1/sg'(s)),

which we want to be nonnegative, and where we have let s=(2kt+r^2)^1/2. 

As a conclusion, we have proved the following.



[Radial solutions of the Cauchy problem (<ref>)] If g:[0,+∞)→  is twice differentiable on (0,+∞) and such that

    g”(s)-1/s g'(s)≥ 0, ∀ s>0,

then the solution of the Cauchy problem (<ref>) starting from u_0(x)=g(| x|) is 

    u(t,x)=g(√(2kt+| x| ^2)).



In other words, in the above situation, the so-called Heat equation (<ref>) does not diffuse but transports.  Let us investigate a few examples, for which we always assume μ >0 and β >0.

 Function g(s):=μ e^- s ^2/2k satisfies (<ref>). From (<ref>) we get the solution 

    u(t,x)=μ e^-te^-| x| ^2/2k.

Notice that u(t,x)=e^-tv(x) where v(x):=μ e^-| x| ^2/2k
is an eigenelement for operator _k^-: as noticed in <cit.>, v solves _k^- v +v=0.















 Function g(s):=μ/(+s^2)^β, >0, satisfies (<ref>). From (<ref>) we recover the solution (<ref>).



Any function g:[0,+∞)→, twice differentiable on (0,+∞), which is nonincreasing and convex satisfies (<ref>). In this framework g(s)=μ e^-s provides the solution u(t,x)=μ e^-√(2kt+| x| ^2) whereas g(s)=μ/(+s)^β, > 0, provides the solution u(t,x)=μ/(+√(2kt+| x| ^2))^β.


The appearance of a transport equation implies very striking phenomena for a so-called Heat equation: as shown by the following example, global extinction in finite time, or quenching, may occur.

[Quenching] Straightforward computations show that the smooth function

    g(s):=e^1/s-1    if  0≤ s<1
    
    0     if  s≥ 1

does satisfy (<ref>). Since u_0(x)=g(| x|) is compactly supported in the ball of radius 1, the associated solution (<ref>) of the Cauchy problem vanishes everywhere as soon as t≥1/2k, that is a quenching phenomena in finite time occurs.




 §.§ Operator ^+_k

Under assumption (<ref>), we have

    ^+_ku=∑ _i=N-k+1^Nλ_i(D^2u)=∂_rrψ+k-1/| x|∂ _r ψ

and thus the Heat equation (<ref>) is transferred into  the linear convection diffusion equation

    ∂ _tψ=∂_rrψ+k-1/r∂ _r ψ.




























We assume that g is bounded on [0,+∞). We denote by g its radial extension to ^k, namely g(x):=g(| x|) for x∈^k. We thus select

    ψ(t,r)=1/(4π t)^k/2∫ _^k e^-| r  -y| ^2/4tg(y)dy=(G_k(t,·)* g)(r ),    t>0,   r∈,

where  is any unit vector in ^k. Since (<ref>) corresponds to solving the radial Heat equation in ^k, the restriction of ψ(t,r) to the t>0, r>0, solves (<ref>) and starts from g(r).
















As a conclusion, we have proved the following.

[Radial solutions of the Cauchy problem (<ref>)] If g:[0,+∞)→ is bounded and such that ψ(t,r) given by (<ref>) satisfies 

    ∂ _rrψ(t,r)≥1/r∂ _r ψ(t,r),  for all  t>0, r>0,

then the solution of the Cauchy problem (<ref>) starting from u_0(x)=g(| x|) is

    u(t,x)=ψ(t,| x|)=1/(4π t)^k/2∫ _ ^k e^-|(| x| -y)| ^2/4t  g(| y|)dy,    t>0,  x∈^N,

where  is any unit vector in ^k.


Let us make a few comments. First, notice that x lives in ^N but we integrate over y∈^k. Next, observe that  (<ref>) does not provide a convolution formula for any radial solution, which would be in contrast with the fact that the equation is fully nonlinear. Actually, (<ref>) provides a convolution formula under condition (<ref>) on the initial data, which is more consistent. Nonetheless, notice that (<ref>) is stable by linear combination with nonnegative coefficients.



For the Gaussian initial data g(s)=1/(4π a)^k/2e^-s^2/4a, a>0, the convolution (<ref>) is straightforwardly computed as ψ(t,r)=1/(4π(a+t))^k/2e^-r^2/4(a+t) which satisfies (<ref>). Hence we get the solution 

    u(t,x)=1/(4π(a+t))^k/2e^-| x|^2/4(a+t)

for t>0, x∈^N. Notice that, for any 1≤ p≤ +∞,

    ‖ u(t,·)‖_L^p(^N)∼C/t^1/2(k-N/p),    as  t→ +∞.

where C=C(a,p,N,k)>0. In particular the L^∞ norm decreases like 1/t^k/2.



 For the step function g(s):=1_(0,1)(s) and k=1 (for simplicity), we use  (<ref>) to compute (∂ _rrψ -1/r∂ _r ψ)(t,r) for t>0, r>0, and observe that it has the sign of 











    B(t,r):=∫ _-1^1  (r(r-y)^2-2ty)
    e^-(r-y)^2/4tdy.

Using a formal calculation software, we get

    B(t,r)=2te^-1+r^2/4t(e^r/2t(-r+r^2+2t)-e^-r/2t(r+r^2+2t)),

which fails to be nonnegative as soon as 0<r<1, 0<t<r-r^2/2. Hence (<ref>) is not satisfied. Notice however that g∉UC(^N) so that the well-posedness of the Cauchy problem is not obvious.


Now, we intend to provide examples of compactly supported initial data that satisfy (<ref>). This is more complicated than checking condition (<ref>), which is the counterpart of condition (<ref>) for  the Heat equation involving _k ^-, since the solution ψ(t,r)  is now given by a convolution, namely (<ref>). With additional assumptions on the initial data, we now try to find a local” sufficient condition for (<ref>) to hold. Assuming that there is >0 such that

    g is of the class C^2 on [0,], nonnegative, g'(0)=0, 
    and g≡ 0 on [,+∞),

the following computations are licit.  Formula (<ref>) yields

    (4π t)^k/2ψ(t,r)=∫_ ^ke^-| re_1-y| ^2/4tg(| y|)dy=∫_| y| <e^-| re_1-y| ^2/4tg(| y|)dy,

where e_1 denotes the first vector of the canonical basis of ^k. In the sequel a generic y∈^k is recast y=(y_1,y') with y_1∈, y'∈^k-1. We differentiate with respect to r and get, using  the shortcut z=z(y):=re_1-y,

    (4π t)^k/2ψ _r (t,r)   =   ∫_| y| <-2(r-y_1)/4t
    e^-| z | ^2/4tg(| y|)dy=-∫ _| y| <∂/∂ y_1[e^-| z| ^2/4t]g(| y|)dy
       =   ∫_| y| <e^-| z| ^2/4ty_1/| y|g'(| y|)dy,

using integration by part over y_1, noticing that the boundary terms vanishes since g()=0. Again we differentiate with respect to r, write ∫_| y| <fdy=∫_| y'| <∫_-α^αfdy_1dy' with the shortcut α=α(y'):=√( ^2-| y'| ^2), use integration by part over y_1  and reach

    (4π t)^k/2ψ _rr (t,r)=∫_| y'| < e^-| y'| ^2/4t-α g'()/(e^-(r-α)^2/4t+e^-(r+α)^2/4t)dy'
    
    
    +∫_| y| <e^-| z| ^2/4t(
    1/| y|g'(| y|)-y_1^2/| y|^3g'(| y|)+y_1^2/| y |^2g”(| y|)
    )dy.

 Notice that the first integral term, over | y'| <, is the boundary term. Putting all together we see that the sign of ψ_rr(t,r)-1/rψ_r(t,r) is that of

    I(t,r):=∫_| y| <e^-| z| ^2/4ty_1^2/| y |^2(g”(| y|)-1/| y|g'(| y|))dy+∫_| y| <e^-| z| ^2/4tr-y_1/r| y|g'(| y|)
    dy
    
    +∫_| y'| < e^-| y'| ^2/4t-α g'()/(e^-(r-α)^2/4t+e^-(r+α)^2/4t)dy'=:(I_1+I_2+I_3)(t,r).

Using again integration by part over y_1 we get

    I_2(t,r)=∫_| y'| <e^-| y'| ^2/4t2t/r(e^-(r-α)^2/4tg'()/-e^-(r+α)^2/4tg'()/)dy'
    
    -2t/r∫_| y| <e^-| z|^2/4ty_1/| y| ^2(g”(| y|)-1/| y|g'(| y|))dy.

Putting all together we arrive at

    I(t,r)   =   ∫_| y| <e^-| z|^2/4ty_1(y_1-2t/r)/| y| ^2(g”(| y|)-1/| y|g'(| y|))dy
          +∫ _| y'| <
    e^-| y'| ^2/4tt/r-g'()/e^-(r+α)^2/4t(e^α r t^-1(α r t^-1-2)+α r t^-1+2
    )dy'
       =:   (J_1+J_2)(t,r).

We easily see that e^λ (λ -2)+λ +2≥ 0 for all λ≥ 0 and, since g'()≤ 0, we have J_2(t,r)≥ 0 for all t>0, r>0. Nonetheless even if we assume

    g”(s)-1/sg'(s)≥ 0,  for all 0< s< ,

we cannot hope the term J_1(t,r) to remain nonegative for all t>0, r>0 — unless it vanishes— because of the term y_1(y_1-2t/r). This is a strong indication that the nonnegative initial data for which (<ref>) holds are rather rare” or, in other words and roughly speaking, condition (<ref>) seems to be very unstable”. In particular for k=1 or g'()=0, the nonnegative favorable term J_2(t,r) vanishes.

Nevertheless,  assuming equality in (<ref>) obviously saves the day and provides the following example, which is an important tool for the proof of Theorem <ref> on the Fujta blow-up phenomena.
 
 Let >0 be given. Function g(s):=( ^2-s^2)_+ clearly satisfies (<ref>) and the equality in (<ref>), so that the solution of the Cauchy problem (<ref>) starting from u_0(x)=g(| x|) is given by the convolution formula (<ref>).



From Example <ref>, Example <ref> and the comparison principle we deduce that, for any nonnegative and nontrivial initial data u_0 (not necessarily radial) having tails that can be dominated by a Gaussian tail, the solution u(t,x) of the Heat equation (<ref>) starting from u_0 satisfies

    C_1/(1+t)^k/2≤‖ u(t,·)‖ _L^∞(^N)≤C_2/(1+t)^k/2,  for all  t≥ 0,

for some positive constants C_1=C_1(u_0), C_2=C_2(u_0).




Assume that the initial data u_0(x)=g(| x|) is such that the conclusion of Theorem <ref> holds, and that g_i:=∫_^k| x| ^i g(| x|)dx<+∞ for i=0,1. Then the solution  becomes asymptotically self-similar in the sense that,

    u(t,·)-g_01/(4π t)^k/2e^-| · | ^2/4t_L^∞( ^N)=o(t^-k/2)  as t→+∞.
 
This can be proved from the convolution formula (<ref>) by reproducing the standard argument for the (classical) Heat equation, see  for instance  the monograph of Giga, Giga and Saal <cit.>.





§ GLOBAL VS BLOW-UP SOLUTIONS FOR THE DOUBLY NONLINEAR CAUCHY PROBLEMS


In this section, as explained in Section <ref>, we wonder if the local solution to the Cauchy problem associated with equations (<ref>) or (<ref>) is global or not. 

Let us recall that, in his seminal work <cit.>, Fujita considered solutions u(t,x) to the nonlinear (p>0) Heat equation 

    ∂ _t u=Δ u+u^1+p   in  (0,∞)×^N,

supplemented with a nonnegative and nontrivial initial data and proved the following: when 0<p<2/N, any  solution blows up in finite time whereas, when p>2/N some solutions with small initial data are global in time. Hence, for equation (<ref>), p_F:=2/N is the so-called Fujita exponent. Let us observe that, as well-known, solutions to the Heat equation  ∂ _t u=Δ u tend to zero as t→∞ like  𝒪(t^-N/2), which is a formal  argument to guess p_F=2/N.

In the sequel we prove that p_F=0 for equation (<ref>) involving _k ^-, whereas p_F=2/k for equation (<ref>) involving _k^+.



 §.§ Operator _k^-


As seen in Example <ref>, the L^∞ norm of some solutions to the Heat equation (<ref>) decrease exponentially fast to zero at large times. This is a strong indication that the Fujita exponent is p_F=0. 

 Let p>0 be given. Assume 0≤ u_0(x)≤ Ce^-| x| ^2/2k for some 0< C≤ 1. Then the solution to (<ref>) starting from u_0 is global in time and satisfies

    ‖ u(t,·)‖_L^∞(^N)≤C/(1-C^p)^1/pe^-t    if  0<C<1
    
    C     if  C=1.




 We define v(t,x):=Ce^-te^-| x| ^2/2k which is the solution of the Heat equation (<ref>) starting from v_0(x)=Ce^-| x| ^2/2k. We look after a supersolution to (<ref>) in the form 
 
    u(t,x)=f(t)v(t,x),

 with f to be chosen and starting from f(0)=1. We compute
 
    (∂ _t u - _k^-u -u^1+p)(t,x)=f'(t)v(t,x)-f^1+p(t)v^1+p(t,x)

 which is nonnegative provided
 
    f'(t)/f^1+p(t)≥‖ v(t,·)‖ _L^∞^p=C^pe^-pt.

 Since C≤ 1 the Cauchy problem f'(t)/f^1+p(t)=C^pe^-pt, f(0)=1 is globally solved as 
 
    f(t)=1/(1+C^p(e^-pt-1))^1/p≤1/(1-C^p)^1/p    if  0<C<1
    
     e^t    if  C=1.

 From the comparison principle, we deduce 0≤ u(t,x)≤u(t,x) for all t>0, x∈^N, which provides the result.


The solutions (<ref>) to the Heat equation (<ref>) provide examples of global solutions to (<ref>) with initial heavy tails, provided p is large enough.

 Let β>0 be given. Let p>1/β be given. Assume 0≤ u_0(x)≤C/(| x|^2+)^β for some C>0, >0 satisfying

    C^p/^pβ-1≤ 2kpβ-1/p.

Then the solution to (<ref>) starting from u_0 is global in time and satisfies

    ‖ u(t,·)‖_L^∞(^N)≤C'/t^β    if C^p/^pβ-1< 2kpβ-1/p
    C'/t^1/p    if C^p/^pβ-1= 2kpβ-1/p,

for some C'=C'(β,p,k,,C)>0.


 We define v(t,x):=C/(| x| ^2+2kt+)^β which is the solution of the Heat equation starting from v_0(x)=C/(| x| ^2+)^β. Next, the proof is similar as the previous one.


From any of the two above propositions, we thus conclude that we do have p_F=0. Notice also that p_F=0 also follows from the following observation  from <cit.>: for any p>0, equation (<ref>) admits the stationary solutions

    (2k/p(μ+| x| ^2))^1/p,

which corresponds to the critical case p=1/β of the above proposition.



 §.§ Operator _k^+


As seen in Example <ref>, the L^∞ norm of some solutions to the Heat equation (<ref>) decrease like t^-k/2 at large times. This is an indication that the Fujita exponent  p_F is smaller than 2/k. This is confirmed by the following construction of global solutions when p>2/k.

 Assume p>2/k. Let a>0 be given.  Assume 0≤ u_0(x)≤C/(4π a)^k/2e^-| x| ^2/4a for some C>0. Then, if C>0 is small enough, the solution to (<ref>) starting from u_0 is global in time and satisfies

    ‖ u(t,·)‖_L^∞(^N)≤C'/(a+t)^k/2

for some C'=C'(p,k,a,C)>0.



 We define v(t,x):=C/(4π(a+t))^k/2e^-| x|^2/4(a+t) which is the solution of the Heat equation (<ref>) starting from v_0(x)=C/(4π a)^k/2e^-| x|^2/4a. We look after a supersolution to (<ref>) in the form 
 
    u(t,x)=f(t)v(t,x),

 with f to be chosen and starting from f(0)=1. We compute
 
    (∂ _t u - _k^+u -u^1+p)(t,x)=f'(t)v(t,x)-f^1+p(t)v^1+p(t,x)

 which is nonnegative provided
 
    f'(t)/f^1+p(t)≥‖ v(t,·)‖ _L^∞^p=C^p/(4π(a+t))^pk/2.

If C>0 is sufficiently small, the Cauchy problem f'(t)/f^1+p(t)=C^p/(4π(a+t))^pk/2, f(0)=1 is globally solved as 
 
    f(t)=1/(1+pC^p/(4π)^pk/2(pk/2-1)(1/(a+t)^pk/2-1-1/a^pk/2-1))^1/p≤1/(1-pC^p/(4π)^pk/2(pk/2-1)a^pk/2-1)^1/p.

 From the comparison principle, we deduce 0≤ u(t,x)≤u(t,x) for all t>0, x∈^N, which provides the result.


Our last main result shows that p_F=2/k.

[Systematic blow-up when p<2/k] Assume 0<p<2/k. Then for any u_0∈ UC(^N) nonnegative and nontrivial, the solution to (<ref>) starting from u_0 blows up in finite time.


 Since the equation is invariant by translation in space and in view of the comparison principle, it is enough to consider the case of the compactly supported initial data

    u_0(x)=g(| x|):=( ^2-| x| ^2)_+,    x∈^N,

for a arbitrary small >0. We assume that the solution u(t,x) (t>0, x∈^N)  to (<ref>) starting from u_0 is global in time and look after a contradiction. To start with, we make the additional assumption (to be removed in the end of the proof) that the viscosity solution is radial and smooth, in the sense that u(t,x)=φ(t,| x|) for some φ=φ(t,r) smooth on (0,+∞)×[0,+∞).

In some related proofs of blow-up phenomena, see <cit.>, <cit.>, <cit.>, the fundamental solution of the underlying linear Heat equation is used. We are not equipped with such a tool but it turns out that the solution of Example <ref> has enough good properties for a modification of the argument to apply. Hence, we denote by v(t,x) (t>0, x∈ ^N) the solution to (<ref>) starting from u_0, as provided by Theorem <ref> and Example <ref>. In particular we have v(t,x)=ψ(t,| x|) for  ψ=ψ(t,r) provided by the convolution formula (<ref>) and smooth on (0,+∞)×[0,+∞).


We define the quantity (notice that we integrate over z∈^k)

    f(t):=∫ _ ^kv(t,| z| e_1)u_0(| z| e_1)dz=∫ _ ^kψ(t,| z|)g(| z|)dz,

where e_1 is the first unit vector of the canonical basis of ^N. We aim at finding estimates of f(t) from below and above which are incompatible as t→+∞. 

From the expression of the initial data, we have

    f(t)≥^2/2∫_| z| </√(2)v(t,| z| e_1)dz.

Since v(t,x) is given by the convolution formula (<ref>) we see, from the expression of the initial data, that, for any | x| </√(2) and t≥ 1, v(t,x)≥C/t^k/2 for some C=C()>0. As a result, we reach the estimate from below

    f(t)≥C_1/t^k/2,   ∀ t≥  1,

for some C_1=C_1()>0.

Next, for a given t>0 and any small α>0, we let

    g(s):=∫_^k v(t-s+α,| z| e_1)u(s,| z| e_1)dz,    0≤ s≤ t.

We differentiate with respect to s and use the equations satisfied by v and u to reach

    g'(s)   =   ∫_ ^k(- _k^+v(t-s+α,| z| e_1) u(s,| z| e_1)+v(t-s+α,| z| e_1) _k ^+u(s,| z| e_1))dz
          +∫_ ^kv(t-s+α,| z| e_1)u^1+p(s,| z| e_1)dz=:h_1(s)+h_2(s).

A first key point is that, as understood in Section <ref> and roughly speaking, _k^+v corresponds to the Laplacian in dimension k<N. Another crucial point is that, for a radial fonction u, 𝒫_k^+u is always larger than the Laplacian in dimension k<N, this following from the beginning of Section <ref>. Precisely, denoting | S_k-1| the area of the unit hypersphere of ^k, we have

    h_1(s)   ≥   | S_k-1|∫_0^+∞((-ψ_rr-k-1/rψ_r)(t-s+α,r)φ(s,r)
          + ψ(s,r)(φ_rr+k-1/rφ_r)(t-s+α,r))r^k-1dr

which is nonnegative as seen by integrating by parts. Next, from the convolution formula (<ref>) and Fubini-Tonelli theorem, we see that, for all τ>0,

    ∫ _^k v(τ,| z| e_1)dz=‖ g(|·|)‖ _L^1( ^k)=:C(,k)^1/p=C^1/p>0.

Therefore we have, from Jensen inequality,

    g'(s)≥ h_2(s)≥ C( ∫ _ ^k v(t-s+α,| z| e_1)u(s,| z| e_1) dz)^1+p=Cg^1+p(s).

Integrating this differential inequality from 0 to t, we get Ct≤-1/p(1/g^p(t)-1/g^p(0))≤1/pg^p(0). Now letting α→ 0, this is recast

    f(t)≤C_2/t^1/p,   ∀ t≥ 1,

for some C_2=C_2(p,k,)>0. As announced, letting t→ +∞ into (<ref>) and (<ref>) contradicts 0<p<2/k.

It remains to remove the assumption that u is radial and smooth, which can be done thanks to the comparison principle and the crucial point mentioned above concerning radial solutions. Indeed, let us denote by w(t,x) the solution to

    ∂ _t w=Δ w +w^1+p   in  (0,+∞)× ^k,

starting from w_0(x)=u_0(x)=g(| x|), for which we know that w(t,x)=θ(t,| x|) for some θ=θ(t,r) smooth on (0,+∞)× [0,+∞). We switch to ^N by letting

    w(t,x):=θ(t,| x| e_1),    t>0,  x∈ ^N.

Since Δw≤𝒫 _k ^+w, we deduce from the comparison principle that w≤ u, and it suffices to prove the blow-up of w. Since w possesses all the necessary properties, we can reproduce the above argument with w playing the role of u. This concludes the proof of Theorem <ref>.


siam  



