Admissible initial growth for diffusion equations with weakly superlinear absorption

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weakly superlinear absorption

Andrey Shishkov, Laurent Véron

To cite this version:

Andrey Shishkov, Laurent Véron. Admissible initial growth for diffusion equations with weakly su- perlinear absorption. Communications in Contemporary Mathematics, World Scientific Publishing, 2015, 18 (5), pp.13. �hal-01136836v3�

with weakly superlinear absorption

Andrey Shishkov

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg str. 74, 83114 Donetsk, Ukraine

Laurent V´eron

Laboratoire de Math´ematiques et Physique Th´eorique, CNRS UMR 6083, Universit´e Fran¸cois-Rabelais, 37200 Tours, France

2010 Mathematics Subject Classification. 35K58; 35K61.

Key words. semilinear heat equations; absorption; maximal and minimal solutions; prospective minimal solution;

non-uniqueness; maximal growth.

AbstractWe study the admissible growth at infinity of initial data of positive solutions of∂tu−

∆u+f(u) = 0 inR+×R^N whenf(u) is a continuous function,mildlysuperlinear at infinity, the model case beingf(u) =uln^α(1 +u) with 1< α <2. We prove in particular that if the growth of the initial data at infinity is too strong, there is no more diffusion and the corresponding solution satisfies the ODE problem∂tφ+f(φ) = 0 onR+ withφ(0) =∞.

Contents

1 Introduction and formulation of the results 1

2 The maximal solution 6

2.1 Proof of Theorem A . . . 6 2.2 Proof of Theorem B . . . 7

3 The prospective minimal solution 8

3.1 The stationary problem . . . 8 3.2 Proof of Theorem C . . . 9

1 Introduction and formulation of the results

Let h be a continuous nondecreasing function defined on R+ and vanishing only at 0.

It is well known that for any continuous and bounded function g belonging to C_b⁺(R^N),

1

the cone of bounded nonnegative continuous functions on R^N, there exists a unique weak solution u:=ug ∈C_b⁺(R+×R^N) of

∂_tu−∆u+uh(u) = 0 in Q^∞

R^N :=R+×R^N,

limt→0u(t, .) =g locally uniformly inR^N. (1.1) Furthermore, the solution u satisfies

0≤u(x, t)≤Φkgk_L∞(t) ∀(t, x)∈Q^∞

R^N, (1.2)

where Φa is the solution of the following Cauchy problem:

Φ_t+ Φh(Φ) = 0 on R+,

Φ(0) =a. (1.3)

Since h(0) = 0 it follows easily that

Φa(t)>0 ∀t >0,∀a>0.

Moreover the family{Φ_a(t)}is monotonically increasing with respect to the parameter a, and the condition:

Z ∞

c

ds

sh(s) <∞, c= const>0, (1.4) is equivalent to the existence of a solution Φ∞(t) of equation in (1.3) such that Φ∞(t)<∞,

∀t >0 and with infinite initial data, i. e.

limt→0Φ∞(t) =∞.

Now we come to our main subject, to study problem (1.1) with an initial data g(x) unbounded and tending to infinity at infinity. It is clear that the character of growth of h(s) at infinity defines the class of initial functions g of solvability of problem under consideration. For example, ifh(s) is bounded, then the corresponding class of solvability is the Tikhonov class [7] {g : g(x) 6cexp(c₁|x|²), c, c₁ = const<∞}. When h(s) tends to infinity at infinity, the class of admissible initial data is larger that the Tikhonov class.

If h(s) increases at infinity fast enough in the sense that condition (1.4) is satisfied, then problem (1.1) is solvable for any nonnegative continuous functiongin the sense that there exists a prospective minimal solution u_g which is the limit when n→ ∞ of the solutions u_gn of

∂tu−∆u+uh(u) = 0 in Q^∞

R^N

t→0limu(t, .) =gχ_Bn inL¹(R^N), (1.5) whereB_ndenotes the open ball of radius nandχ_A is characteristic function of the setA, and there holds

0≤u_g ≤Φ∞. (1.6)

But the main question is to know whether the prospective minimal solution is truly a solution with initial datag(.). If it is the case we say that the prospective minimal solution is the minimal solution. Another assumption on h which plays a fundamental role in the study is the so-called Keller-Osserman condition (see [1], [6]),

Z ∞

a

ds

pH(s) <∞ for some a >0, where H(t) = Z _s

0

th(t)dt. (1.7) When this condition is satisfied, for any R >0 there exist solutions of

−∆u+uh(u) = 0 in B_R

|x|→Rlim u(x) =∞. (1.8) This condition gives rise to a localization phenomenon thanks to which we prove an existence and uniqueness result of the solution with initial datag.

Theorem AAssume r7→rh(r) is convex and satisfies(1.7). Then for any g∈C⁺(R^N), u_g is the minimal solution with initial data g. Furthermore it is the unique nonnegative solution of problem (1.1).

In the class of functionsh(s) of the formh(s) =sln^α(1+s), condition (1.4) is equivalent to α > 1, but condition (1.7) is equivalent to α > 2. In general it is easy to show that condition (1.7) is stronger than condition (1.4) .

When h(s) is a power function, the class of existence and uniqueness is much larger than the class of Th. A. A complete description of this existence and uniqueness class is based upon the notion of initial trace which has been thoroughly investigated by Marcus and V´eron [3], [4] and Gkikas and V´eron [2].

When Z ∞

a

ds

pH(s) =∞ ∀a >0, (1.9) uniqueness may not hold in the class of unbounded solutions. If, for anyb >0,V_b denotes the maximal solution of the following Cauchy problem

Vrr+N −1

r Vr−V h(V) = 0 on (0, Rb) V(0) =b

V_r(0) = 0,

(1.10)

then Rb =∞. Actually, multiplying (1.10) by Vr, we get easily 2⁻¹ d

dr|V_r|² = d

drH(V)−N−1

r |V_r|² 6 d

drH(V).

Since Vr(0) = 0, we derive

Vr(r)6

√ 2p

H(V(r)) ∀r >0.

Integrating this last inequality we obtain the a prioriestimate V(R) =V_b(R)6V¯_b(R) ∀R >0, where the function ¯V_b(R) is defined by the identity:

Fb( ¯Vb(R)) =

√

2R ∀R >0, where Fb(v) :=

Z v b

ds pH(s),

(see e.g. [8] also). Moreover it is easy to see that for arbitrary a > b >0, V_a(r)> V_b(r)

∀r >0. Actually, due to the monotonicity ofh, there holds V_arr(0) = 1

NV_a(0)h(V_a(0)) = 1

Nah(a)> 1

Nbh(b) =V_brr(0),

from (1.10). SinceVar(0) =Vbr(0) = 0 it follows from this last inequality that the function r 7→ W(r) = V_a(r)−V_b(r) is increasing near r = 0; it remains increasing on whole R+, since, if we assume that there exists r₀ > 0 where W reaches a local maximum, then Wr(r0) = 0, Wrr(r0)≤0, but from equation (1.10) we have:

W_rr(r₀) =V_a(r₀)h(V_a(r₀))−V_b(r₀)h(V_b(r₀))>0,

which is a contradiction. Furthermore, Nguyen Phuoc and V´eron proved in [5] that if g satisfies

Vc(|x|)≤g(x)≤V_b(|x|) ∀x∈R^N, (1.11) for some b > c > 0, then there exists at least two different solutions of (1.1) defined in Q^∞

R^N: the minimal one u_g which satisfies

u_g(x, t)≤Φ∞(t) ∀(x, t)∈Q^∞

R^N, (1.12)

and another one ug such that

V_c(|x|)≤u_g(x, t)≤V_b(|x|) ∀(x, t)∈Q^∞

R^N. (1.13)

It is not clear wether there exists a maximal solution or not. However, if g satisfies (1.11), then there exists a minimal solution u_g,c,b and a maximal one u_g,c,b in the class Ec,b(g) of solutions of problem (1.1), satisfying inequalities (1.13). These two solutions can be constructed by the following approximate scheme: we define the sequence{u_n} of solutions of the Cauchy-Dirichlet problem

∂tu−∆u+uh(u) = 0 in Q^∞_Bn :=R+×Bn

u(t, x) =Vc(n) in ∂`Q^∞_Bn :=R+×∂Bn

u(0, .) =g inB_n;

(1.14)

then it is easy to check using comparison principle that the sequence {u_n} is increasing and converges to u_g,c,b. Similarly, the sequence {u_n} of solutions of the same equation in

Q^∞_B

n with the same initial data and boundary value V_b(n) is decreasing and converges to ug,c,b.

In this paper we consider the case where the initial dataggrows at infinity faster than any function Vb with arbitrary b <∞. Our aim is to describe analogs of the ”maximal”

solution u_g from (1.13) and prospective minimal solution{u_g} from (1.5), (1.6). For any a >0 we denote byu:=u_a,n the solution of

∂_tu−∆u+uh(u) = 0 in Q^∞_B

n :=R+×B_n u(t, x) =V_a(n) in ∂_`Q^∞_B

n :=R+×∂B_n u(0, .) = min{V_a, g} inB_n,

(1.15)

Due to the comparison principle it is clear that ua,n 6 Va in Q^∞_Bn. The next result highlights a phenomenon of instantaneous blow-up of the maximal solution if the initial data grows too fast at infinity.

Theorem B Assume r 7→rh(r) is convex and satisfies (1.4) and (1.9). If g ∈C⁺(R^N), satisfies

|x|→∞lim g(x)

Va(|x|) =∞ ∀a >0, (1.16)

then for arbitrary m ∈ N the sequence {u_a,n}_n>m decreases and converges in Q^∞_B

m to a functionua which is solution of(1.1)with initial data min{V_a, g}. Furthermoreua(t, x)→

∞ for any (t, x)∈Q^∞

R^N as a→ ∞. Thus, the function identically equal to ∞ inQ^∞

R^N can be considered as the ”maximal” solution of problem (1.1)in the case of (1.16).

Let us remark that in subsection 3.1 we find the asymptotic expression of the functions V_a for the model nonlinearities h, which makes the condition (1.16) more explicit.

A fundamental example of equations with nonlinearities satisfying (1.4) and (1.9) is provided by

∂_tu−∆u+uln^α(1 +u) = 0 in Q^∞

R^N (1.17)

with 1< α≤2. With this specific type of nonlinearity we prove:

Theorem C Assume 1 < α < 2 and g ∈C⁺(R^N), satisfies condition (1.16), which due to Proposition 3.1 has now the following form

lim

|x|→∞g(x) exp

−c_α|x|^2−α2

=∞, c_α=

2−α 2

_2−α² .

Then the prospective minimal solution u_g of (1.17)with initial data g isΦ∞.

Notice that the two types of generalized approximative solutions of problem (1.1), obtained in Theorems B, C, ”forget” the real initial condition from (1.1): in another words, they realize infinite initial jump.

2 The maximal solution

2.1 Proof of Theorem A

The fact that u_g is a solution of (1.1), and clearly the minimal one, is more or less stan- dard, but we recall its proof for the sake of completeness since it contains the localization principle. Form∈N^∗ let vm be the minimal solution of

−∆v+vh(v) = 0 in B_m

|x|→mlim v(x) =∞. (2.1) Such a solution exists by [1] or [6] because (1.7) holds. It is nonnegative and radial as limit of the nonnegative radial functionsv_m,k,k∈N^∗which are the solutions of (2.1) with finite boundary data v_m,k =k on ∂Bm. Moreover v_m,k, and thus vm, is an increasing function of |x|. Clearly v_m≥0 and it is a stationary solution of (1.1) inQ^∞_B

m. For n≥m, let u_gn be the solution of (1.5) andγ_m = max{g(x) :|x| ≤m}. Thenv_m+γ_m is a super solution of (1.5) in Q^∞_Bm which dominates un on ∂_`Q^∞_Bm∪ {0} ×Bm Thusvm+γm ≥un in Q^∞_Bm. The set of functions {u_n} is bounded and uniformly continuous in Q^T_Bm−1, by standard regularity theory for parabolic equations, thus it converges uniformly therein to u_g and u_gb_Bm×{0}=g. This implies thatu_g hasg as initial data.

Assume now thatu another solution with the same initial datag. We setw=u−u_g. Since r7→rh(r) is convex andu−u_g is positive,

uh(u)≥u_gh(u_g) + (u−u_g)h(u−u_g).

Thereforewis a subsolution of problem (1.1), andw(t, x)→0 ast→0, locally uniformly inR^N. By the comparison principle

w(t, x)≤v_n(x) in Q^∞_Bn,

wherevnsatisfies (2.1) inBn. Furthermoren7→vnis decreasing with limitv∞ asn→ ∞.

The function v∞ verifies

−∆v+vh(v) = 0 in R^N.

Furthermore it is nonnegative, radial and nondecreasing with respect to |x|. In order to prove that v= 0, we return to vn which satisfies

vn r=r^1−N Z r

0

s^N−1vn(s)h(vn(s))ds≤vn(r)h(vn(r))r^1−N Z r

0

s^N−1ds= r

Nvn(r)h(vn(r)).

Thus

−v_{n rr}+v_n(r)h(v_n(r)) = N −1 r v_{n r}≤

1− 1

N

v_n(r)h(v_n(r)) which implies

−v_{n rr}+ 1

Nv_n(r)h(v_n(r))≤0.

Integrating twice yields

Z ∞

vn(r)

ds pH(t) ≥

r2

N(n−r), (2.2)

where H has been defined in (1.7). If we had v∞(r)>0 for somer >0, it would imply

∞>

Z ∞

v∞(r)

ds

p2H(t) ≥ ∞,

a contradiction. Thusv∞(r) = 0 and w(t, x) = 0.

2.2 Proof of Theorem B

We recall that (1.16) holds and that u_a,n denotes the solution of (1.15). Since V_ab_Q^∞

Bn is the solution of the Cauchy-Dirichlet problem

∂tu−∆u+uh(u) = 0 in Q^∞_Bn :=R+×Bn

u(t, x) =Va(n) in ∂`Q^∞_Bn :=R+×∂Bn

u(0, .) =V_a in B_n,

(2.3)

it is larger thanua,n. Thusua,n+1b_∂`Q^∞

Bn≤ua,nb_∂`Q^∞

Bn=Va. Sinceua,n(0, .) =ua,n+1b_Bn(0, .) it follows that ua,n+1b_Q^∞

Bn≤ua,n. Then {u_a,n} is a decreasing sequence, and its limit ua

is a solution of (1.1), which the first claim. By the same argument, ua,n≤u_b,n+1b_Q^∞

Bn in Q^∞_Bn for b > a. Hence ua ≤ u_b. We introduce the sequence {r_a} : ra → ∞ as a → ∞ defined by:

r_a= inf{r >0 :g(x)>V_a(x) ∀ |x|>r}, (2.4) and, for n≥r_a, we set w_a,n=V_a−u_a,n. By convexityw_a,n satisfies

∂_tw_a,n−∆w_a,n+w_a,nh(w_a,n)≤0 in Q^∞_B

n :=R+×B_n, w_a,n(t, x) = 0 in ∂_`Q^∞_B

n :=R+×∂B_n, w_a,n(0, x) = (V_a−g)₊ inB_n.

(2.5)

Therefore

wa,n(t, x)<Φ∞(t) in Q^∞_Bn, (2.6) where Φ∞ is defined in (1.3) with a=∞. Actually,

Z ∞

Φ∞(t)

ds

sh(s) =t. (2.7)

Notice also that the sequence {w_a,n} is increasing and it converges, as n → ∞, to wa = Va−ua, which is dominated by Φ∞ Thus

ua(t, x)≥Va(x)−Φ∞(t)≥a−Φ∞(t) in Q^∞

R^N. (2.8)

Letting a→ ∞ implies the claim.

3 The prospective minimal solution

In this section we consider equation (1.17) with 1< α <2.

3.1 The stationary problem

Proposition 3.1 Assume 1< α <2, a >0 and Va is the solution of Vrr+N −1

r Vr−V ln^α(V + 1) = 0 in R+

V_r(0) = 0 V(0) =a.

(3.1)

Then

Va(r) =e^cαr

2−α2 +O(1) as r → ∞, (3.2)

where cα= ^2−α_{2 2−α}² .

Proof. We writeW = ln(V + 1). Since V is increasing W >0, W_r≥0 and W_rr+W_r²+N−1

r W_r−(1−e^−W)W^α= 0 inR+. (3.3) Thus

W_rr+W_r²−(1−e^−W)W^α ≤0.

If we setρ=W and p(ρ) =W_r(r), then ρ∈[a,∞) and pp⁰+p²−(1−e^−ρ)ρ^α ≤0.

This is a linear differential inequality in the unknownp². Integrating yields p²(ρ)≤2e^−2ρ

Z ρ a

(e^2s−e^s)s^αds=ρ^α+O(1). (3.4) Thus Wr(r)≤W^α2(r) +O(1) as r→ ∞ which implies

W(r)≤c_αr^2−α2 +O(1) as r→ ∞. (3.5) Due to (3.5) relation (3.4) yields also the following inequality

0< Wr ≤c

α

α2r^2−αα (1 +o(1)).

Since W(r)→ ∞asr→ ∞, it follows from (3.3) and (3.4) that for any >0 there exists r>0 such that

W_rr+W_r² ≥(1−)W^α on [r,∞).

Integrating this ordinary differential inequality we get

W(r)≥(1−))cαr^2−α2 (1 +o(1)) as r → ∞. (3.6) Since is arbitrary, we derive

W(r) =c_αr^2−α2 (1 +o(1)) as r→ ∞. (3.7) From the above estimates, we can improve (3.6). Using (3.4) and (3.7) we deduce from (3.3):

pp⁰+p² = (1−e^−ρ)ρ^α−N−1

r Wr ≥(1−e^−ρ)ρ^α−cρ^α−1, from which it follows easily

p²(ρ)≥2e^−2ρ Z ρ

a

e^2s(s^α−c⁰s^α−1)ds=ρ^α+O(1), (3.8) by l’Hospital rule. Combined with (3.7) and (3.5), it implies

W(r) =cαr^2−α2 +O(1) as r→ ∞. (3.9) Returning to Va, we derive

Va(r) =e^cαr

2−α2 +O(1) as r → ∞. (3.10)

Remark. Ifα= 2, the same method yields

V_a(r) =e^er+O(1) as r→ ∞. (3.11)

3.2 Proof of Theorem C

We recall that the prospective minimal solution u_g is the limit, when n → ∞ of the (increasing) sequence of solutions{u_g`n}of

∂tu−∆u+uln^α(u+ 1) = 0 inQ^∞

R^N

u(0, .) =gχB<sub>`n</sub> inR<sup>N</sup>, (3.12) where{`_n}is any increasing sequence converging to∞. Furthermore, if we replacegby its maximal radial minorant defined by ˜g(r) := min_|x|=rg(x), it satisfies also (1.16). Because of (1.16) there exists a sequence {r_n}tending to infinity such that

r_n= inf{r >0 : ˜g(s)≥V_n(s) ∀s≥r}, then ˜g(r_n) =V_n(r_n).

Step 1: Estimate from below. Put

gn(|x|) =

min{˜g(rn),g(|x|)}˜ if|x|< rn

˜

g(rn) if|x| ≥rn. Let u_gn be the minimal solution of

∂tu−∆u+uln^α(u+ 1) = 0 inQ^∞

R^N

u(0, .) =gn inR^N. (3.13)

Thenu_gn ≤Φ∞. For any sequence{`<sub>k</sub>}converging to infinity and any fixedk, there exists n<sub>k</sub> such that for n≥n<sub>k</sub>, there holdsgχ<sub>B`k</sub> ≤g_n. Since the sequence {u_g

n} is increasing, its limitu∞is a solution of (1.3) inQ^R_∞^N which is larger thanug<sub>`k</sub> for any`_k, and therefore larger also than u_˜g. However, since g_n≤˜g,u∞≤u_g˜. This implies

u∞=u_˜g6Φ∞. (3.14)

Next, since u_gn(0, x) ≤g(rn), it follows that u_gn(t, x) ≤ g(rn). Let ωn = Φ_g(rn), i.e.

the solution of (1.3) witha=g(rn), then ωn satisfies Z g(rn)

ωn(t)

ds sh(s) =t, and u_gn ≥wn where wn is the minimal solution of

∂_tw−∆w+wln^α(ω_n+ 1) = 0 inQ^∞

R^N

w(0, .) =gn inR^N. (3.15) If we setwn(t, x) =e^{−R0tlnα(ωn(s)+1)ds}zn(t, x), then

∂tzn−∆zn= 0 inQ^∞

R^N

zn(0, .) =gn inR^N. (3.16)

Since

zn(t, x) = 1 (4πt)^N2

Z

R^N

e^−|x−y|

2

4t gn(y)dy, we can writewn(t, x) =In(t, x) +Jn(t, x) where

In(t, x) = e^{−R0tlnα(ωn(s)+1)ds} (4πt)^N2

Z

|y|≤rn

e^−|x−y|

2

4t gn(y)dy, (3.17) and

Jn(t, x) = e^{−R0tlnα(ωn(s)+1)ds}g(r˜ n) (4πt)^N2

Z

|y|>rn

e^−|x−y|

2

4t dy. (3.18)

Clearly

Jn(t, x)≥ e^{−R0tlnα(ωn(s)+1)ds}˜g(rn) (4πt)^N2

Z

|y|>rn+|x|

e^−|y|

2 4t dy

≥ e^{−R0tlnα(ωn(s)+1)ds}˜g(r_n) (4πt)^N2

Z

|z|>rn+|x|

e^−z

2 4tdz

!N

.

(3.19)

This integral term can be estimated by introducing Gauss error function ercf(x) = 2

√π Z ∞

x

e^−z2dz. (3.20)

In dimension N, it implies easily

Jn(t, x)≥e^{−R0tlnα(ωn(s)+1)ds}˜g(rn)

ercf

rn+|x|

2√ t

N

. (3.21)

Since

ercf(x) = e^−x2 x√

t(1 +O(x⁻²)) as x→ ∞, we derive

J_n(t, x)≥ ˜g(rn) ((rn+|x|)²t)^N2

e^{−R0tlnα(ωn(s)+1)ds−N(rn+|x|)24t}

1 +O

t r²_n

. (3.22)

We write ˜g(r) = exp(γ(r))−1 and set A_n(t, x) =γ(r_n)−

Z t 0

ln^α(ω_n(s) + 1)ds−N(r_n+|x|)²

4t −Nln(r_n+|x|)−N 2 lnt.

In order to have an estimate on ωn(s), we fix t≤ 1 and ˜g(rn) ≥ 1. There exists a0 ≥ 1 such that

min

ω_a(t)

ωa(t) + 1 : 0≤t≤1, a≥a₀

≥ 1 2. In such a range ofa andt,

ω⁰+ωln^α(ω+ 1) =ω⁰+ ω

ω+ 1(ω+ 1) ln^α(ω+ 1)

≥ω⁰+1

2(ω+ 1) ln^α(ω+ 1), which yields

ln^α(ωn(s) + 1)≤

2γ^α−1(rn) 2 + (α−1)sγ^α−1(r_n)

_α−1^α . From this inequality, we derive

Z t 0

ln^α(ω_n(s) + 1)ds≤ Z t

0

2γ^α−1(r_n) 2 + (α−1)sγ^α−1(r_n)

_α−1^α ds

≤2^α−1α γ(rn)

Z tγ^α−1(rn) 0

(2 + (α−1)τ)^−α−1α dτ.

Therefore

An(t, x)≥γ(rn)− N(rn+|x|)²

4t −Nln(rn+|x|)−N 2 lnt

−2^α−1α γ(rn)

Z tγ^α−1(rn) 0

(2 + (α−1)τ)^−α−1α dτ.

(3.23)

Step 2: The maximal admissible growth. We claim that lim inf

|x|→∞ |x|^−2−α2 ln ˜g(|x|)> N^2−α1 =⇒ lim

n→∞u_gn(t, x) = Φ∞(t) ∀(t, x)∈Q^∞

R^N. (3.24) By replacing τ 7→(2 + (α−1)τ)^−α−1α by its maximal value on (0, tγ^α−1(rn)),

2^α−1α γ(r_n)

Z tγ^α−1(rn) 0

(2 + (α−1)τ)^−α−1α dτ ≤γ^α(r_n)t.

Then

A_n(t, x)≥γ(r_n)−N(r_n+|x|)²

4t −Nln(r_n+|x|)−N

2 lnt−γ^α(r_n)t:=B_n(t, x), (3.25) and

∂tBn(t, x) = N(rn+|x|)² 4t² −N

2t −γ^α(rn).

Thus

∂_tB_n(t, x) = 0 andt >0⇐⇒t:=t_n= N(r_n+|x|)² N+p

N²+ 4N(r_n+|x|)²γ^α(r_n). (3.26) ThereforeA_n(t_n, x) is bounded from below by the maximum ofB_n(t, x) which is achieved fort=t_n and

B_n(t_n, x) =γ(r_n)−Nln(r_n+|x|)−N +p

N²+ 4N(r_n+|x|)²γ^α(r_n) 4

− N(r_n+|x|)²γ^α(r_n) N +p

N²+ 4N(rn+|x|)²γ^α(rn) −N

2 ln N(r_n+|x|)² N+p

N²+ 4N(rn+|x|)²γ^α(rn)

! .

Since r_n→ ∞ asn→ ∞it follows from last representation that Bn(tn, x) =rnγ^α2(rn) γ^1−α2(rn)

rn

−N¹²(1 +νn(x))

!

, (3.27)

where ν_n(x)→0 asn→ ∞uniformly on any compact set in R^N. Therefore ifg satisfies lim inf

|x|→∞|x|^−2−α2 ln ˜g(|x|)> N^2−α1 , (3.28)

then there holds

Jn(tn, x)

n→∞

→ ∞=⇒ lim

tn→0u_gn(tn, x) =∞, (3.29) uniformly on compact subsets of R^N. We fix m >0, denote by λ_m the first eigenvalue of

−∆ in H₀¹(Bm), with corresponding eigenfunction φm normalized by sup_Bmφm = 1 and set, for >0,

Wm,(t, x) =e^−(t+)λmΦ∞(t+)φm(x) ∀(t, x)∈Q^B_∞^m. Then

∂_tW_m,−∆W_m,+W_m,ln^α(W_m,+ 1) =W_m,

ln^α(W_m,+ 1)−ln^α(Φ∞(t+) + 1)

≤0.

Since u_gn increases to the prospective minimal solution u_˜g, it follows due to (3.29 ) that there existsn such that

u_˜g(tn, x)≥u_gn(tn, x)≥Wm,(tn+, x) ∀x∈Bm. Last inequality in virtue of comparison principle implies

u_g(t, x)≥W_m,(t+, x) ∀(t, x)∈Q^B_∞^m, t≥t_n.

Letting → 0 yields u_g ≥ Wm,0 in Q^B_∞^m. Since limm→∞φm(x) = 1, uniformly on any compact subset ofR^N and limm→∞λ_m= 0 we deriveu_˜g >Φ∞and finallyu_g >Φ∞. This

inequality together with (3.14 ) leads to u= Φ∞.

Remark. In the caseα= 2, there holds Z t

0

ln²(ω_n(s) + 1)ds≤4γ(r_n)

Z tγ(rn) 0

(2 +τ)⁻²dτ ≤tγ(r_n). (3.30) Therefore (3.25) is replaced by

A_n(t, x)≥γ(r_n)−tγ²(r_n)− N(r_n+|x|)²

4t −Nln(r_n+|x|)− N

2 lnt:=B_n(t, x). (3.31) A similarly, there exists tn>0 where t7→Bn(t, x) is maximum and in that case

B_n(t_n, x) =γ(r_n)−Nln(r_n+|x|)−N +p

N²+ 4N(rn+|x|)²γ²(rn) 4

− N(rn+|x|)²γ²(rn) N +p

N²+ 4N(rn+|x|)²γ²(rn) −N

2 ln N(rn+|x|)² N +p

N²+ 4N(rn+|x|)²γ²(rn)

! ,

which yields

Bn(tn, x) =γ(rn)−rnγ(rn)(N¹² −νn(x)), (3.32) where νn(x)→0 as n→ ∞ uniformly on any compact set inR^N. Thus Bn(tn, x)→ −∞

as n → ∞. A similar type of computation shows that the expression I_n(t, x) defined in (3.17) converges to 0, whatever is the sequence {r_n}converging to ∞.

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