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Revisiting Taibleson's theorem

  • A new characterization of the weighted Taibleson's theorem for generalized Hölder spaces is given via a Hadamard-Liouville type operator (Djrbashian's generalized fractional operator).

    Citation: Humberto Rafeiro, Joel E. Restrepo. Revisiting Taibleson's theorem[J]. Electronic Research Archive, 2022, 30(2): 565-573. doi: 10.3934/era.2022029

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  • A new characterization of the weighted Taibleson's theorem for generalized Hölder spaces is given via a Hadamard-Liouville type operator (Djrbashian's generalized fractional operator).



    We consider the Cauchy problem for the nonlinear damped wave equation with linear damping

    uttΔu+ut=|u|p+|u|q+w(x),t>0,xRN, (1.1)

    where N1, p,q>1, wL1loc(RN), w0 and w0. Namely, we are concerned with the existence and nonexistence of global weak solutions to (1.1). We mention below some motivations for studying problems of type (1.1).

    Consider the semilinear damped wave equation

    uttΔu+ut=|u|p,t>0,xRN (1.2)

    under the initial conditions

    (u(0,x),ut(0,x))=(u0(x),u1(x)),xRN. (1.3)

    In [12], a Fujita-type result was obtained for the problem (1.2) and (1.3). Namely, it was shown that,

    (ⅰ) if 1<p<1+2N and RNuj(x)dx>0, j=0,1, then the problem (1.2) and (1.3) admits no global solution;

    (ⅱ) if 1+2N<p<N for N3, and 1+2N<p< for N{1,2}, then the problem (1.2) and (1.3) admits a unique global solution for suitable initial values.

    The proof of part (ⅰ) makes use of the fundamental solution of the operator (ttΔ+t)k. In [15], it was shown that the exponent 1+2N belongs to the blow-up case (ⅰ). Notice that 1+2N is also the Fujita critical exponent for the semilinear heat equation utΔu=|u|p (see [3]). In [6], the authors considered the problem

    utt+(1)m|x|αΔmu+ut=f(t,x)|u|p+w(t,x),t>0,xRN (1.4)

    under the initial conditions (1.3), where m is a positive natural number, α0, and f(t,x)0 is a given function satisfying a certain condition. Using the test function method (see e.g. [11]), sufficient conditions for the nonexistence of a global weak solution to the problem (1.4) and (1.3) are obtained. Notice that in [6], the influence of the inhomogeneous term w(t,x) on the critical behavior of the problem (1.4) and (1.3) was not investigated. Recently, in [5], the authors investigated the inhomogeneous problem

    uttΔu+ut=|u|p+w(x),t>0,xRN, (1.5)

    where wL1loc(RN) and w0. It was shown that in the case N3, the critical exponent for the problem (1.5) jumps from 1+2N (the critical exponent for the problem (1.2)) to the bigger exponent 1+2N2. Notice that a similar phenomenon was observed for the inhomogeneous semilnear heat equation utΔu=|u|p+w(x) (see [14]). For other results related to the blow-up of solutions to damped wave equations, see, for example [1,2,4,7,8,9,13] and the references therein.

    In this paper, motivated by the above mentioned works, our aim is to study the effect of the gradient term |u|q on the critical behavior of problem (1.5). We first obtain the Fujita critical exponent for the problem (1.1). Next, we determine its second critical exponent in the sense of Lee and Ni [10].

    Before stating the main results, let us provide the notion of solutions to problem (1.1). Let Q=(0,)×RN. We denote by C2c(Q) the space of C2 real valued functions compactly supported in Q.

    Definition 1.1. We say that u=u(t,x) is a global weak solution to (1.1), if

    (u,u)Lploc(Q)×Lqloc(Q)

    and

    Q(|u|p+|u|q)φdxdt+Qw(x)φdxdt=QuφttdxdtQuΔφdxdtQuφtdxdt, (1.6)

    for all φC2c(Q).

    The first main result is the following Fujita-type theorem.

    Theorem 1.1. Let N1, p,q>1, wL1loc(RN), w0 and w0.

    (i) If N{1,2}, for all p,q>1, problem (1.1) admits no global weak solution.

    (ii) Let N3. If

    1<p<1+2N2or1<q<1+1N1,

    then problem (1.1) admits no global weak solution.

    (iii) Let N3. If

    p>1+2N2andq>1+1N1,

    then problem (1.1) admits global solutions for some w>0.

    We mention below some remarks related to Theorem 1.1. For N3, let

    pc(N,q)={1+2N2ifq>1+1N1,ifq<1+1N1.

    From Theorem 1.1, if 1<p<pc(N,q), then problem (1.1) admits no global weak solution, while if p>pc(N,q), global weak solutions exist for some w>0. This shows that pc(N,q) is the Fujita critical exponent for the problem (1.1). Notice that in the case N3 and q>1+1N1, pc(N,q) is also the critical exponent for uttΔu=|u|p+w(x) (see [5]). This shows that in the range q>1+1N1, the gradient term |u|q has no influence on the critical behavior of uttΔu=|u|p+w(x).

    It is interesting to note that the gradient term |u|q induces an interesting phenomenon of discontinuity of the Fujita critical exponent pc(N,q) jumping from 1+2N2 to as q reaches the value 1+1N1 from above.

    Next, for N3 and σ<N, we introduce the sets

    I+σ={wC(RN):w0,|x|σ=O(w(x)) as |x|}

    and

    Iσ={wC(RN):w>0,w(x)=O(|x|σ) as |x|}.

    The next result provides the second critical exponent for the problem (1.1) in the sense of Lee and Ni [10].

    Theorem 1.2. Let N3, p>1+2N2 and q>1+1N1.

    (i) If

    σ<max{2pp1,qq1}

    and wI+σ, then problem (1.1) admits no global weak solution.

    (ii) If

    σmax{2pp1,qq1},

    then problem (1.1) admits global solutions for some wIσ.

    From Theorem 1.2, if N3, p>1+2N2 and q>1+1N1, then problem (1.1) admits a second critical exponent, namely

    σ=max{2pp1,qq1}.

    The rest of the paper is organized as follows. In Section 2, some preliminary estimates are provided. Section 3 is devoted to the study of the Fujita critical exponent for the problem (1.1). Namely, Theorem 1.1 is proved. In Section 4, we study the second critical exponent for the problem (1.1) in the sense of Lee and Ni. Namely, we prove Theorem 1.2.

    Consider two cut-off functions f,gC([0,)) satisfying

    f0,f0,supp(f)(0,1)

    and

    0g1,g(s)={1if0s1,0ifs2.

    For T>0, we introduce the function

    ξT(t,x)=f(tT)g(|x|2T2ρ)=F(t)G(x),(t,x)Q, (2.1)

    where 2 and ρ>0 are constants to be chosen. It can be easily seen that

    ξT0andξTC2c(Q).

    Throughout this paper, the letter C denotes various positive constants depending only on known quantities.

    Lemma 2.1. Let κ>1 and >2κκ1. Then

    Q|ξT|1κ1|ΔξT|κκ1dxdtCT2ρκκ1+Nρ+1. (2.2)

    Proof. By (2.1), one has

    Q|ξT|1κ1|ΔξT|κκ1dxdt=(0F(t)dt)(RN|G|1κ1|ΔG|κκ1dx). (2.3)

    Using the properties of the cut-off function f, one obtains

    0F(t)dt=0f(tT)dt=T0f(tT)dt=T10f(s)ds,

    which shows that

    0F(t)dt=CT. (2.4)

    Next, using the properties of the cut-off function g, one obtains

    RN|G|1κ1|ΔG|κκ1dx=Tρ<|x|<2Tρ|G|1κ1|ΔG|κκ1dx (2.5)

    On the other hand, for Tρ<|x|<2Tρ, an elementary calculation yields

    ΔG(x)=Δ[g(r2T2ρ)],r=|x|=(d2dr2+N1rddr)g(r2T2ρ)=2T2ρg(r2T2ρ)2θ(x),

    where

    θ(x)=Ng(r2T2ρ)g(r2T2ρ)+2(1)T2ρr2g(r2T2ρ)g(r2T2ρ)2+2T2ρr2g(r2T2ρ)g(r2T2ρ).

    Notice that since Tρ<r<2Tρ, one deduces that

    |θ(x)|C.

    Hence, it holds that

    |ΔG(x)|CT2ρg(|x|2T2ρ)2,Tρ<|x|<2Tρ

    and

    |G(x)|1κ1|ΔG(x)|κκ1CT2ρκκ1g(|x|2T2ρ)2κκ1,Tρ<|x|<2Tρ.

    Therefore, by (2.5) and using the change of variable x=Tρy, one obtains

    RN|G|1κ1|ΔG|κκ1dxCT2ρκκ1Tρ<|x|<2Tρg(|x|2T2ρ)2κκ1dx=CT2ρκκ1+Nρ1<|y|<2g(|y|2)2κκ1dy,

    which yields (notice that >2κκ1)

    RN|G|1κ1|ΔG|κκ1dxCT2ρκκ1+Nρ. (2.6)

    Finally, using (2.3), (2.4) and (2.6), (2.2) follows.

    Lemma 2.2. Let κ>1 and >κκ1. Then

    Q|ξT|1κ1|ξT|κκ1dxdtCTρκκ1+Nρ+1. (2.7)

    Proof. By (2.1) and the properties of the cut-off function g, one has

    Q|ξT|1κ1|ξT|κκ1dxdt=(0F(t)dt)(Tρ<|x|<2Tρ|G|1κ1|G|κκ1dx). (2.8)

    On the other hand, for Tρ<|x|<2Tρ, an elementary calculation shows that

    |G(x)|=2T2ρ|x||g(|x|2T2ρ)|g(|x|2T2ρ)122Tρ|g(|x|2T2ρ)|g(|x|2T2ρ)1CTρg(|x|2T2ρ)1,

    which yields

    |G|1κ1|G|κκ1CTρκκ1g(|x|2T2ρ)κκ1,Tρ<|x|<2Tρ.

    Therefore, using the change of variable x=Tρy and the fact that >κκ1, one obtains

    Tρ<|x|<2Tρ|G|1κ1|G|κκ1dxCTρκκ1Tρ<|x|<2Tρg(|x|2T2ρ)κκ1dx=CTρκκ1+Nρ1<|y|<2g(|y|2)κκ1dy,

    i.e.

    Tρ<|x|<2Tρ|G|1κ1|G|κκ1dxCTρκκ1+Nρ. (2.9)

    Using (2.4), (2.8) and (2.9), (2.7) follows.

    Lemma 2.3. Let κ>1 and >2κκ1. Then

    Q|ξT|1κ1|ξTtt|κκ1dxdtCT2κκ1+1+Nρ. (2.10)

    Proof. By (2.1) and the properties of the cut-off functions f and g, one has

    Q|ξT|1κ1|ξTtt|κκ1dxdt=(T0F(t)1κ1|F(t)|κκ1dt)(0<|x|<2TρG(x)dx). (2.11)

    An elementary calculation shows that

    F(t)1κ1|F(t)|κκ1CT2κκ1f(tT)2κκ1,0<t<T,

    which yields (since >2κκ1)

    T0F(t)1κ1|F(t)|κκ1dtCT2κκ1+1. (2.12)

    On the other hand, using the change of variable x=Tρy, one obtains

    0<|x|<2TρG(x)dx=0<|x|<2Tρg(|x|2T2ρ)dx=TNρ0<|y|<2g(|y|2)dy,

    which yields

    0<|x|<2TρG(x)dxCTNρ. (2.13)

    Therefore, using (2.11), (2.12) and (2.13), (2.10) follows.

    Lemma 2.4. Let κ>1 and >κκ1. Then

    Q|ξT|1κ1|ξTt|κκ1dxdtCTκκ1+1+Nρ. (2.14)

    Proof. By (2.1) and the properties of the cut-off functions f and g, one has

    Q|ξT|1κ1|ξTt|κκ1dxdt=(T0F(t)1κ1|F(t)|κκ1dt)(0<|x|<2TρG(x)dx). (2.15)

    An elementary calculation shows that

    F(t)1κ1|F(t)|κκ1CTκκ1f(tT)κκ1,0<t<T,

    which yields (since >κκ1)

    T0F(t)1κ1|F(t)|κκ1dtCT1κκ1. (2.16)

    Therefore, using (2.13), (2.15) and (2.16), (2.14) follows.

    Proposition 3.1. Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). Then there exists a constant C>0 such that

    Qw(x)φdxdtCmin{A(φ),B(φ)}, (3.1)

    for all φC2c(Q), φ0, where

    A(φ)=Qφ1p1|φtt|pp1dxdt+Qφ1p1|φt|pp1dxdt+Qφ1p1|Δφ|pp1dxdt

    and

    B(φ)=Qφ1p1|φtt|pp1dxdt+Qφ1p1|φt|pp1dxdt+Qφ1q1|φ|qq1dxdt.

    Proof. Let u be a global weak solution to problem (1.1) with

    (u,u)Lploc(Q)×Lqloc(Q).

    Let φC2c(Q), φ0. Using (1.6), one obtains

    Q|u|pφdxdt+Qw(x)φdxdtQ|u||φtt|dxdt+Q|u||Δφ|dxdt+Q|u||φt|dxdt. (3.2)

    On the other hand, by ε-Young inequality, 0<ε<13, one has

    Q|u||φtt|dxdt=Q(|u|φ1p)(φ1p|φtt|)dxdtεQ|u|pφdxdt+CQφ1p1|φtt|pp1dxdt. (3.3)

    Here and below, C denotes a positive generic constant, whose value may change from line to line. Similarly, one has

    Q|u||Δφ|dxdtεQ|u|pφdxdt+CQφ1p1|Δφ|pp1dxdt (3.4)

    and

    Q|u||φt|dxdtεQ|u|pφdxdt+CQφ1p1|φt|pp1dxdt (3.5)

    Therefore, using (3.2), (3.3), (3.4) and (3.5), it holds that

    (13ε)Q|u|pφdxdt+Qw(x)φdxdtCA(φ).

    Since 13ε>0, one deduces that

    Qw(x)φdxdtCA(φ). (3.6)

    Again, by (1.6), one has

    Q(|u|p+|u|q)φdxdt+Qw(x)φdxdt=QuφttdxdtQuΔφdxdtQuφtdxdt=Quφttdxdt+QuφdxdtQuφtdxdt,

    where denotes the inner product in RN. Hence, it holds that

    Q(|u|p+|u|q)φdxdt+Qw(x)φdxdtQ|u||φtt|dxdt+Q|u||φ|dxdt+Q|u||φt|dxdt. (3.7)

    By ε-Young inequality (ε=12), one obtains

    Q|u||φtt|dxdt12Q|u|pφdxdt+CQφ1p1|φtt|pp1dxdt, (3.8)
    Q|u||φt|dxdt12Q|u|pφdxdt+CQφ1p1|φt|pp1dxdt, (3.9)
    Q|u||φ|dxdt12Q|u|qφdxdt+CQφ1q1|φ|qq1dxdt. (3.10)

    Next, using (3.7), (3.8), (3.9) and (3.10), one deduces that

    12Q|u|qφdxdt+Qw(x)φdxdtCB(φ),

    which yields

    Qw(x)φdxdtCB(φ). (3.11)

    Finally, combining (3.6) with (3.11), (3.1) follows.

    Proposition 3.2. Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). Then

    Qw(x)ξTdxdtCmin{A(ξT),B(ξT)}, (3.12)

    for all T>0, where ξT is defined by (2.1).

    Proof. Since for all T>0, ξTC2c(Q), ξT0, taking φ=ξT in (3.1), (3.12) follows.

    Proposition 3.3. Let wL1loc(RN), w0 and w0. Then, for sufficiently large T,

    Qw(x)ξTdxdtCT, (3.13)

    where ξT is defined by (2.1).

    Proof. By (2.1) and the properties of the cut-off functions f and g, one has

    Qw(x)ξTdxdt=(T0f(tT)dt)(0<|x|<2Tρw(x)g(|x|2T2ρ)dx). (3.14)

    On the other hand,

    T0f(tT)dt=T10f(s)ds. (3.15)

    Moreover, for sufficiently large T (since w,g0),

    0<|x|<2Tρw(x)g(|x|2T2ρ)dx0<|x|<1w(x)g(|x|2T2ρ)dx. (3.16)

    Notice that since wL1loc(RN), w0, w0, and using the properties of the cut-off function g, by the domianted convergence theorem, one has

    limT0<|x|<1w(x)g(|x|2T2ρ)dx=0<|x|<1w(x)dx>0.

    Hence, for sufficiently large T,

    0<|x|<1w(x)g(|x|2T2ρ)dxC. (3.17)

    Using (3.14), (3.15), (3.16) and (3.17), (3.13) follows.

    Now, we are ready to prove the Fujita-type result given by Theorem 1.1. The proof is by contradiction and makes use of the nonlinear capacity method, which was developed by Mitidieri and Pohozaev (see e.g. [11]).

    Proof of Theorem 1.1. (ⅰ) Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). By Propositions 3.2 and 3.3, for sufficiently large T, one has

    CTmin{A(ξT),B(ξT)}, (3.18)

    where ξT is defined by (2.1),

    A(ξT)=Qξ1p1T|ξTtt|pp1dxdt+Qξ1p1T|ξTt|pp1dxdt+Qξ1p1T|ΔξT|pp1dxdt (3.19)

    and

    B(ξT)=Qφ1p1|ξTtt|pp1dxdt+Qξ1p1T|ξTt|pp1dxdt+Qξ1q1T|ξT|qq1dxdt. (3.20)

    Taking >max{2pp1,qq1}, using Lemmas 2.1, 2.3, 2.4 with κ=p, and Lemma 2.2 with κ=q, one obtains the following estimates

    Qξ1p1T|ξTtt|pp1dxdtCT2pp1+1+Nρ, (3.21)
    Qξ1p1T|ξTt|pp1dxdtCTpp1+1+Nρ, (3.22)
    Qξ1p1T|ΔξT|pp1dxdtCT2ρpp1+Nρ+1, (3.23)
    Qξ1q1T|ξT|qq1dxdtCTρqq1+Nρ+1. (3.24)

    Hence, by (3.19), (3.21), (3.22) and (3.23), one deduces that

    A(ξT)C(T2pp1+1+Nρ+Tpp1+1+Nρ+T2ρpp1+Nρ+1).

    Observe that

    2pp1+1+Nρ<pp1+1+Nρ. (3.25)

    So, for sufficiently large T, one deduces that

    A(ξT)C(Tpp1+1+Nρ+T2ρpp1+Nρ+1).

    Notice that the above estimate holds for all ρ>0. In particular, when ρ=12, it holds that

    A(ξT)CTpp1+1+N2 (3.26)

    Next, by (3.20), (3.21), (3.22), (3.24) and (3.25), one deduces that

    B(ξT)C(Tpp1+1+Nρ+Tρqq1+Nρ+1).

    Similarly, the above inequality holds for all ρ>0. In particular, when ρ=p(q1)q(p1), it holds that

    B(ξT)CTpp1+1+Np(q1)q(p1). (3.27)

    Therefore, it follows from (3.18), (3.26) and (3.27) that

    0<Cmin{Tpp1+N2,Tpp1+Np(q1)q(p1)},

    which yields

    0<CTpp1+N2:=Tα(N) (3.28)

    and

    0<CTpp1+Np(q1)q(p1):=Tβ(N). (3.29)

    Notice that for N{1,2}, one has

    α(N)={p12(p1)ifN=1,1p1ifN=2,

    which shows that

    α(N)<0,N{1,2}.

    Hence, for N{1,2}, passing to the limit as T in (3.28), a contradiction follows (0<C0). This proves part (ⅰ) of Theorem 1.1.

    (ⅱ) Let N3. In this case, one has

    α(N)<0p<1+2N2.

    Hence, if p<1+2N2, passing to the limit as T in (3.28), a contradiction follows. Furthermore, one has

    β(N)<0q<1+1N1.

    Hence, if q<1+1N1, passing to the limit as T in (3.29), a contradiction follows. Therefore, we proved part (ⅱ) of Theorem 1.1.

    (ⅲ) Let

    p>1+2N2andq>1+1N1. (3.30)

    Let

    u(x)=ε(1+r2)δ,r=|x|,xRN, (3.31)

    where

    max{1p1,2q2(q1)}<δ<N22 (3.32)

    and

    0<ε<min{1,(2δ(N2δ2)1+2qδq)1min{p,q}1}. (3.33)

    Note that due to (3.30), the set of δ satisfying (3.32) is nonempty. Let

    w(x)=Δu|u|p|u|q,xRN.

    Elementary calculations yield

    w(x)=2δε[(1+r2)δ12r2(δ+1)(1+r2)δ2+(N1)(1+r2)δ1]εp(1+r2)δp2qδqεqrq(1+r2)(δ+1)q. (3.34)

    Hence, one obtains

    w(x)2δε[(1+r2)δ12(δ+1)(1+r2)δ1+(N1)(1+r2)δ1]εp(1+r2)δp2qδqεq(1+r2)(δ+1)q+q2=2δε(N2δ2)(1+r2)δ1εp(1+r2)δp2qεqδq(1+r2)(δ+12)q.

    Next, using (3.32) and (3.33), one deduces that

    w(x)ε[2δ(N2δ2)εp12qεq1δq](1+r2)δ1>0.

    Therefore, for any δ and ε satisfying respectively (3.32) and (3.33), the function u defined by (3.31) is a stationary solution (then global solution) to (1.1) for some w>0. This proves part (ⅲ) of Theorem 1.1.

    Proposition 4.1. Let N3, σ<N and wI+σ. Then, for sufficiently large T,

    Qw(x)ξTdxdtCTρ(Nσ)+1. (4.1)

    where ξT is defined by (2.1).

    Proof. By (3.14) and (3.15), one has

    Qw(x)ξTdxdt=CT0<|x|<2Tρw(x)g(|x|2T2ρ)dx. (4.2)

    On the other hand, by definition of I+σ, and using that g(s)=1, 0s1, for sufficiently large T, one obtains (since w,g0)

    0<|x|<2Tρw(x)g(|x|2T2ρ)dx0<|x|<Tρw(x)g(|x|2T2ρ)dx=0<|x|<Tρw(x)dxTρ2<|x|<Tρw(x)dxCTρ2<|x|<Tρ|x|σdx=CTρ(Nσ).

    Hence, using (4.2), (4.1) follows.

    Now, we are ready to prove the new critical behavior for the problem (1.1) stated by Theorem 1.2.

    Proof of Theorem 1.2. (ⅰ) Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). By Propositions 3.2 and 4.1, for sufficiently large T, one obtains

    CTρ(Nσ)+1min{A(ξT),B(ξT)}, (4.3)

    where ξT, A(ξT) and B(ξT) are defined respectively by (2.1), (3.19) and (3.20). Next, using (3.26) and (4.3) with ρ=12, one deduces that

    0<CTpp1+σ2. (4.4)

    Observe that

    pp1+σ2<0σ<2pp1.

    Hence, if σ<2pp1, passing to the limit as T in (4.4), a contradiction follows. Furthermore, using (3.27) and (4.3) with ρ=p(q1)q(p1), one deduces that

    0<CTpp1(σ(q1)q1). (4.5)

    Observe that

    pp1(σ(q1)q1)<0σ<qq1.

    Hence, if σ<qq1, passing to the limit as T in (4.5), a contradiction follows. Therefore, part (ⅰ) of Theorem 1.2 is proved.

    (ⅱ) Let

    σmax{2pp1,qq1}. (4.6)

    Let u be the function defined by (3.31), where

    σ22<δ<N22 (4.7)

    and ε satisfies (3.33). Notice that since σ<N, the set of δ satisfying (4.7) is nonempty. Moreover, due to (4.6) and (4.7), δ satisfies also (3.32). Hence, from the proof of part (iii) of Theorem 1.2, one deduces that

    w(x)=Δu|u|p|u|q>0,xRN.

    On the other hand, using (3.34) and (4.7), for |x| large enough, one obtains

    w(x)C(1+|x|2)δ1C|x|2δ2C|x|δ,

    which proves that wIσ. This proves part (ⅱ) of Theorem 1.2.

    We investigated the large-time behavior of solutions to the nonlinear damped wave equation (1.1). In the case when N{1,2}, we proved that for all p>1, problem (1.1) admits no global weak solution (in the sense of Definition 1). Notice that from [5], the same result holds for the problem without gradient term, namely problem (1.5). This shows that in the case N{1,2}, the nonlinearity |u|q has no influence on the critical behavior of problem (1.5). In the case when N3, we proved that, if 1<p<1+2N2 or 1<q<1+1N1, then problem (1.1) admits no global weak solution, while if p>1+2N2 and q>1+1N1, global solutions exist for some w>0. This shows that in this case, the Fujita critical exponent for the problem (1.1) is given by

    pc(N,q)={1+2N2ifq>1+1N1,ifq<1+1N1.

    From this result, one observes two facts. First, in the range q>1+1N1, from [5], the critical exponent pc(N,q) is also equal to the critical exponent for the problem without gradient term, which means that in this range of q, the nonlinearity |u|q has no influence on the critical behavior of problem (1.5). Secondly, one observes that the gradient term induces an interesting phenomenon of discontinuity of the Fujita critical exponent pc(N,q) jumping from 1+2N2 to as q reaches the value 1+1N1 from above. In the same case N3, we determined also the second critical exponent for the problem (1.1) in the sense of Lee and Ni [10], when p>1+2N2 and q>1+1N1. Namely, we proved that in this case, if σ<max{2pp1,qq1} and wI+σ, then there is no global weak solution, while if max{2pp1,qq1}σ<N, global solutions exist for some wIσ. This shows that the second critical exponent for the problem (1.1) in the sense of Lee and Ni is given by

    σ=max{2pp1,qq1}.

    We end this section with the following open questions:

    (Q1). Find the first and second critical exponents for the system of damped wave equations with mixed nonlinearities

    {uttΔu+ut=|v|p1+|v|q1+w1(x)in(0,)×RN,vttΔv+vt=|u|p2+|u|q2+w2(x)in(0,)×RN,

    where pi,qi>1, wiL1loc(RN), wi0 and wi0, i=1,2.

    (Q2). Find the Fujita critical exponent for the problem (1.1) with w0.

    (Q3) Find the Fujita critical exponent for the problem

    uttΔu+ut=1Γ(1α)t0(ts)α|u(s,x)|pds+|u|q+w(x),t>0,xRN,

    where N1, p,q>1, 0<α<1 and wL1loc(RN), w0. Notice that in the limit case α1, the above equation reduces to (1.1).

    (Q4) Same question as above for the problems

    uttΔu+ut=|u|p+1Γ(1α)t0(ts)α|u(s,x)|qds+w(x)

    and

    uttΔu+ut=1Γ(1α)t0(ts)α|u(s,x)|pds+1Γ(1β)t0(ts)β|u(s,x)|q+w(x),

    where 0<α,β<1.

    The author is supported by Researchers Supporting Project RSP-2020/4, King Saud University, Saudi Arabia, Riyadh.

    The author declares no conflict of interest.



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