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Well-posedness, asymptotic stability and blow-up results for a nonlocal singular viscoelastic problem with logarithmic nonlinearity

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Abstract

Considered herein is the well-posedness, asymptotic stability and blow-up of the initial-boundary value problem for nonlocal singular viscoelastic wave equation with logarithmic nonlinearity \(u_{tt}-\frac{1}{x}(x u_{x})_x-\frac{1}{x}(x u_{xt})_x+\int \limits _{0}^{t}m(t-\lambda )\frac{1}{x}(x u_{x}(x, \lambda ))_x \hbox {d}\lambda =|u|^{r-2}u\ln |u|\) subject to a nonlocal boundary condition. Through the effective combining of Galerkin approximation method, modified potential well theory, perturbed energy method, convexity theory and differential-integral inequality techniques, we firstly demonstrate the global existence and uniqueness of weak solutions in certain weighted Sobolev spaces; Secondly, we establish the explicit polynomial and exponential energy decay estimates under some suitable conditions; Finally, we investigate the finite time blow-up criterion and then derive its upper and lower bounds of blow-up time. The above conclusions extend and improve some results in the literatures.

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References

  1. Ionkin, N.I.: The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition. Differ. Uravn. 13(2), 294–304 (1977)

    MathSciNet  Google Scholar 

  2. Kamynin, L.I.: A boundary-value problem in the theory of heat conduction with a nonbclassical boundary condition. UZhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 4(6), 1006–1024 (1964)

    Google Scholar 

  3. Cahlon, B., Shi, K.P.: Stepwise stability for the heat equation with a nonlocal constraint. SIAM J. Numer. Anal. 32(2), 571–593 (1995)

    MathSciNet  Google Scholar 

  4. Cannon, J.R.: The solution of the heat equation subject to the specification of energy. Q. Appl. Math., 21, 155–160 (1963)

  5. Choi, Y.S., Chan, K.-Y.: A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Anal.: Theory Methods Appl. 18(4), 317–331 (1992)

  6. Wu, B., Yu, J., Wang, Z.: Uniqueness and stability of an inverse kernel problem for type \(III\) thermoelasticity. J. Math. Anal. Appl. 402(1), 242–254 (2013)

    MathSciNet  Google Scholar 

  7. Ewing, R.E., Lin, T.: A class of parameter estimation techniques for fluid flow in porous media. Adv. Water Resour. 14(2), 89–97 (1991)

    MathSciNet  Google Scholar 

  8. Di, H.F., Song, Z.F.: Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms. Opuscula Math. 42(2), 119–155 (2022)

    MathSciNet  Google Scholar 

  9. Berrimi, S., Messaoudi, S.A.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64(10), 2314–2331 (2006)

    MathSciNet  Google Scholar 

  10. Kim, J.A., Han, Y.H.: Blow up of solutions of a nonlinear viscoelastic wave equation. Acta Appl. Math. 111(1), 1–6 (2009)

    MathSciNet  Google Scholar 

  11. Song, H.T., Zhong, C.K.: Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal. Real World Appl. 11, 3877–3883 (2010)

    MathSciNet  Google Scholar 

  12. Song, H.T., Xue, D.S.: Blow up in a nonlinear viscoelastic wave equation with strong damping. Nonlinear Anal.: Theory Methods Appl. 109, 245–251 (2014)

    MathSciNet  Google Scholar 

  13. Messaoudi, S.A., Tatar, N.E.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 30, 665–680 (2007)

    MathSciNet  Google Scholar 

  14. Liu, W.J.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal.: Theory Methods Appl. 73(6), 1890–1904 (2010)

    MathSciNet  Google Scholar 

  15. Cavalcanti, M.M., Cavalcanti, V.N.D., Ferreira, J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24(14), 1043–1053 (2001)

    MathSciNet  Google Scholar 

  16. Xu, R.Z., Yang, Y.B., Liu, Y.C.: Global well-posedness for strongly damped viscoelastic wave equation. Appl. Anal. 92(1), 138–157 (2013)

    MathSciNet  Google Scholar 

  17. Cavalcanti, M.M., Oquendo, H.P.: Frictional versus viscoelastic damping in a semi-linear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)

    MathSciNet  Google Scholar 

  18. Cavalcanti, M.M., Cavalcanti, V.N.D., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. (44) (2002)

  19. Cavalcanti, M.M., Cavalcanti, V.N.D., Filho, J.S.P., Soriano, J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integral Equ. 14(1), 85–116 (2001)

    MathSciNet  Google Scholar 

  20. Cavalcanti, M.M., Cavalcanti, V.N.D., Martinez, P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal.: Theory Methods Appl. 68, 177–193 (2008)

    MathSciNet  Google Scholar 

  21. Messaoudi, S.A., Mustafa, M.I.: On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal. Real World Appl. 10(5), 3132–3140 (2009)

    MathSciNet  Google Scholar 

  22. Lu, L.Q., Li, S.J., Chai, S.G.: On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution. Nonlinear Anal. Real World Appl. 12(1), 295–303 (2012)

    MathSciNet  Google Scholar 

  23. Barrow, D., John, P.: Parsons, inflationary models with logarithmic potentials. Phys. Rev. D 52(10), 5576 (1995)

    Google Scholar 

  24. Enqvist, K., McDonald, J.: Q-balls and baryogenesis in the MSSM. Phys. Lett. B 425(3–4), 309–321 (1998)

    Google Scholar 

  25. Górka, P.: Logarithmic Klein–Gordon equation. Acta Phys. Pol. B 40(1), 59 (2009)

    MathSciNet  Google Scholar 

  26. Pata, V., Zelik, S.: Smooth attractors for strongly damped wave equations. Nonlinearity 19(7), 1495–1506 (2006)

    MathSciNet  Google Scholar 

  27. Di, H.F., Shang, Y.D., Song, Z.F.: Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity. Nonlinear Anal. Real World Appl. 51, 102968 (2020)

    MathSciNet  Google Scholar 

  28. Ma, L.W., Fang, Z.B.: Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source. Math. Methods Appl. Sci. 41(7), 2639–2653 (2018)

    MathSciNet  Google Scholar 

  29. Han, X.S.: Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics. Bull. Korean Math. Soc. 50(1), 275–283 (2013)

    MathSciNet  Google Scholar 

  30. Ye, Y.J.: Global solution and blow-up of logarithmic Klein–Gordon equation. Bull. Korean Math. Soc. 57(2), 281–294 (2020)

    MathSciNet  Google Scholar 

  31. Lian, W., Xu, R.Z.: Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. 9(1), 613–632 (2019)

    MathSciNet  Google Scholar 

  32. Hao, J.H., Du, F.Q.: Decay and blow-up for a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity. J. Math. Anal. Appl. 506(1), 125608 (2022)

    MathSciNet  Google Scholar 

  33. Liao, M.L.: The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity. Evol. Equ. Control Theory 11(3), 781–792 (2022)

    MathSciNet  Google Scholar 

  34. Irkl, N., Pikin, E., Agarwal, P.: Global existence and decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with logarithmic nonlinearity. Math. Methods Appl. Sci. 45(5), 2921–2948 (2022)

    MathSciNet  Google Scholar 

  35. Al-Gharabli, M.M., Guesmia, A., Messaoudi, S.: Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity. Commun. Pure Appl. Anal. 18(1), 159–180 (2019)

    MathSciNet  Google Scholar 

  36. Al’shin, A.B., Korpusov, M.O., Siveshnikov, A.G.: Blow up in nonlinear Sobolev type equations. Walter de Gruyter 15, 62–67 (2011)

    MathSciNet  Google Scholar 

  37. Mesloub, S., Messaoudi, S.A.: A nonlocal mixed semilinear problem for second-order hyperbolic equations. Electron. J. Differ. Equ. 30(30), 495–500 (2003)

    MathSciNet  Google Scholar 

  38. Mesloub, S., Bouziani, A.: On a class of singular hyperbolic equation with a weighted integral condition. Int. J. Math. Math. Sci. 22(3), 511–519 (1999)

    MathSciNet  Google Scholar 

  39. Liu, W.J., Sun, Y., Li, G.: On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term. Mathematics 48(1), 1967–1976 (2017)

    MathSciNet  Google Scholar 

  40. Mesloub, S., Messaoudi, S.A.: Global existence, decay, and blow up of solutions of a singular nonlocal viscoelastic problem. Acta Appl. Math. 110(2), 705–724 (2010)

    MathSciNet  Google Scholar 

  41. Wu, S.T.: Blow-up of solutions for a singular nonlocal viscoelastic equation. J. Partial Differ. Equ. 24(2), 140–149 (2011)

    MathSciNet  Google Scholar 

  42. Mecheri, H., Mesloub, S., Messaoudi, S.A.: On solutions of a singular viscoelastic equation with an integral condition. Georgian Math. J. 16(4), 761–778 (2009)

    MathSciNet  Google Scholar 

  43. Levine, H.A.: Some nonexistence and instability theorems for solutions of formally parbabolic equations of the form \(Pu_t= -Au + F(u)\). Arch. Ration. Mech. Anal. 51(5), 371–386 (1973)

    Google Scholar 

  44. Di, H.F., Shang, Y.D.: Existence, nonexistence and decay estimate of global solutions for a viscoelastic wave equation with nonlinear boundary damping and internal source terms. Eur. J. Pure Appl. Math. 10(4), 668–701 (2017)

    MathSciNet  Google Scholar 

  45. Messaoudi, S.A.: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341(2), 1457–1467 (2008)

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    Huafei Di & Yi Qiu

  2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    Huafei Di

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  1. Huafei Di
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Correspondence to Huafei Di.

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This work was completed with the support of Natural Science Foundation of China (No. 11801108), Guangdong Basic and Applied Basic Research Foundation (Nos. 2021A1515010314, 2023A1515030107), Science and Technology Planning Project of Guangzhou City (No. 202201010111), Guangzhou City-College-Eenterprise Joint Funding Project (No. 2024A03J0156), Tertiary Education Scientific Research Project of Guangzhou Municipal Education Bureau (No. 202235103), Guangzhou Education Scientific Research Project (No. 202214066).

Appendix A

Appendix A

For the readers’ convenience, we will provide the details about the proofs of Lemmas 2.4 and 2.5 in this Appendix.

Proof

(Proof of Lemma 2.4.)

  1. (1)

    Applying the definition of J(u), we discover

    $$\begin{aligned} J(\omega u)=\frac{\omega ^{2}}{2}\big (k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})(t)\big )+\frac{\omega ^{r}}{r^{2}}\big (1-r\ln \omega \big )\Vert u\Vert _{r}^{r}-\frac{\omega ^{r}}{r}\int \limits _{0}^{\iota }x|u|^{r}\ln |u|\hbox {d}x. \end{aligned}$$
    (6.1)

    Hence, considering \(\Vert u\Vert _{r}\ne 0\), we can see that (1) holds;

  2. (2)

    By a series of calculations, we have

    $$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u) =\omega \left( k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})\left( t\right) -\omega ^{r-2}\int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x-\omega ^{r-2}\ln \omega \Vert u\Vert _{r}^{r}\right) . \end{aligned}$$
    (6.2)

    Let \(\omega f(\omega )=\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)\), where

    $$\begin{aligned} f(\omega ):=k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})(t)-\omega ^{r-2}\int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x-\omega ^{r-2}\ln \omega \Vert u\Vert _{r}^{r}. \end{aligned}$$

    So we can show that

    $$\begin{aligned} f^{\prime }(\omega )= -\omega ^{r-3}\left[ \left( r-2\right) \int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x+\left( r-2\right) \ln \omega \Vert u\Vert _{r}^{r}+\Vert u\Vert _{r}^{r}\right] . \end{aligned}$$

    If we choose

    $$\begin{aligned} \omega _{1}=\exp \left( \frac{\left( r-2\right) \int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x+\Vert u\Vert _{r}^{r}}{\left( 2-r\right) \Vert u\Vert _{r}^{r}}\right) >0, \end{aligned}$$
    (6.3)

    it follows that \(f^{\prime }(\omega _{1})=0\), \(f^{\prime }(\omega )>0\), for \(0<\omega <\omega _{1}\) and \(f^{\prime }(\omega )<0 \) for \(\omega >\omega _{1}\). Since

    $$\begin{aligned} \lim _{\omega \rightarrow 0}f\left( w\right) =k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})(t)>0, \ \ \ \lim _{\omega \rightarrow \infty }f\left( w\right) =-{\infty }, \end{aligned}$$

    we deduce that there exist exactly one \(\omega _*>0\) such that \(f(\omega _* )=0,\) i.e., \( \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)|_{\omega =\omega _*}=0.\)

  3. (3)

    Since \(\omega f(\omega )=\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)\), we obtain that \(\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)>0\) when \(\omega \in (0,\omega _*)\), \(\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)<0\) when \(\omega \in (\omega _*,\infty )\). Therefore, the conclusion of (3) holds.

  4. (4)

    From \(I(\omega u)=\omega \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u) \) and (3), we can immediately conclude that (4) holds.

\(\square \)

Proof

(Proof of Lemma 2.5.) From (2.6) and definitions of N and d, we know that \(J(u)>0\) for \(u\in N\) and \(d\ge 0\). Now, we will show that there exists a positive function \(u\in N\) and satisfies \(J(u)=d\). In fact, we choose \(\{{u_i}\}_{i=1}^{\infty }\subset N\) to be a minimizing sequence for J. It means that

$$\begin{aligned} \lim _{i\rightarrow \infty }J\left( u_{i}\right) =d. \end{aligned}$$
(6.4)

Thus, we can see that \(\{{|u_i|}\}_{i=1}^{\infty }\subset N\) is also a minimizing sequence for J. Hence, without loss of generality, we suppose \(u_{i}>0\) a.e. in \( (0,\iota ).\) Since \(\{J({u_i})\}_{i=1}^{\infty } \) is bounded and \( I(u_{i})=0\), we obtain from (2.6) that \(\{{u_i}\}_{i=1}^{\infty }\) is bounded in \( H_{0} \)-norm. Here, we pick \(\nu >0 \) small enough such that \(H_{0} \hookrightarrow L_{x}^{r+\nu }\) (\(2<r+\nu <4\)) is compact. Thus, we get that there exists a function u and a subsequence of \(\{u_i\}_{i=1}^\infty \) denoted by \(\{u_{{i}}\}_{i=1}^\infty \) satisfying

$$\begin{aligned} \begin{aligned}&u_{{i}} \rightarrow u \; \text {in } H_{0} \; \text {weakly}, \;{i}\rightarrow \infty , \\&u_{{i}} \rightarrow u \;\text {in} \;L_{x}^{p+\nu }\;\text {strongly}, \; {i}\rightarrow \infty , \\&u_{{i}} \rightarrow u \;\text {a.e.}\; \text {in}\;(0,\iota ),\; {i}\rightarrow \infty . \end{aligned} \end{aligned}$$
(6.5)

Therefore, we get that \(u\ge 0 \) a.e. in \((0,\iota )\).

Next, applying dominated convergence theorem, it follows that

$$\begin{aligned}{} & {} \int \limits _{0}^{\iota }x|u|^{r}\ln |u|\hbox {d}x=\lim _{{i}\rightarrow \infty }\int \limits _{0}^{\iota }x|u_{{i}}|^{r}\ln |u_{{i}}|\hbox {d}x, \end{aligned}$$
(6.6)
$$\begin{aligned}{} & {} \quad \int \limits _{0}^{\iota }x|u|^{r}\hbox {d}x=\lim _{{i}\rightarrow \infty }\int \limits _{0}^{\iota }x|u_{{i}}|^{r}\hbox {d} x. \end{aligned}$$
(6.7)

By the weak lower semi-continuity of \(\Vert \cdot \Vert _{H_{0}}\), we have

$$\begin{aligned} \Vert u\Vert _{H_{0}}\le \liminf _{{i}\rightarrow \infty }\left\| u_{{i}}\right\| _{H_{0}}. \end{aligned}$$
(6.8)

In view of (6.6)–(6.8) and definition of J(u),  there appears the relation

$$\begin{aligned} J(u)\le \liminf _{{i}\rightarrow \infty }J\left( u_{{i}}\right) =d. \end{aligned}$$
(6.9)

Since \(\{u_{i}\}\subset N,\) then we have \(I(u_{i})=0\). Thus, the combination of (6.6)–(6.8) and definition of I(u), we get

$$\begin{aligned} I(u)\le \liminf _{{i}\rightarrow \infty }I\left( u_{{i}}\right) =0. \end{aligned}$$
(6.10)

When \(I(u)=0\), we discover from assumption (A1) that

$$\begin{aligned} \begin{aligned} k\Vert u_{{i}}\Vert _{H_{0}}^{2}&\le k(t)\Vert u_{{i}}\Vert _{H_{0}}^{2} \le \int \limits _{0}^{\iota }x|u_{{i}}|^{r}\ln |u_{{i}}|\hbox {d}x \\&\le \int \limits _{0}^{\iota }x|u_{{i}}|^{r}\left( e\nu \right) ^{-1}|u_{{i}}|^{\nu }d x \le \frac{B_1^{r+\nu }}{e\nu }\left\| u_{{i}}\right\| _{H_{0}}^{r+\nu }, \end{aligned} \end{aligned}$$
(6.11)

where applying the fact \(y^{-\nu }\ln y\le (e\nu )^{-1}\) \((y,\nu >0)\) and Sobolev embedding inequality \(\Vert u_{{i}}\Vert _{r+\nu }\le B_{1}\Vert u_{{i}}\Vert _{H_{0}}.\) The above inequality implies that

$$\begin{aligned} \int \limits _{0}^{\iota }x|u_{{i}}|^{r}\ln |u_{{i}}|\hbox {d}x\ge k\Vert u_{{i}}\Vert _{H_{0}}^{2}\ge k\left( \frac{ke\nu }{B_1^{r+\nu }}\right) ^\frac{2}{r+\nu -2}, \end{aligned}$$
(6.12)

which together (6.6) yields \(\int \limits _{0}^{\iota }x|u|^{r}\ln |u|\hbox {d}x\ge k\left( \frac{ke\nu }{B_1^{r+\nu }}\right) ^\frac{2}{r+\nu -2}>0\). Thus, we conclude that \(u\ne 0\).

If \(I(u)<0\), by the Lemma 2.2 (4), we know that there exist a \(\omega _*\) such that \(0<\omega _*<1\) and \(I(\omega _* u)=0.\) Hence, a series of calculation gives that

$$\begin{aligned} \begin{aligned} d\le J(\omega _{*}u)&=\frac{r-2}{2r}\left( k(t)\Vert \omega _*u\Vert _{H_{0}}^{2}+(m\circ {(\omega _*u_{x}}))(t)\right) +\frac{1}{r^{2}}\Vert \omega _{*}u\Vert _{p}^{p} \\&=\omega _{*}^{2}\left[ \left( \frac{r-2}{2r}\bigg )k(t)\Vert u\Vert _{H_{0}}^{2} +\left( \frac{r-2}{2r}\right) (m\circ {u_{x}})(t)\right) +\frac{1}{r^{2}}\omega _{*}^{r-2}\Vert u\Vert _{r}^{r}\right] \\&\le \omega _{*}^{2}\lim _{{i}\rightarrow \infty }\inf \left\{ \left[ \left( \frac{r-2}{2r}\right) k(t)\Vert u_{{i}}\Vert _{H_{0}}^{2}+\left( \frac{r-2}{2r}\right) (m\circ { u_{xi}})(t)\right] +\frac{1}{r^{2}}\Vert u_{{i}}\Vert _{r}^{r} \right\} \\&=\omega _{*}^{2}\lim _{{i}\rightarrow \infty } \inf J(u_{i})=\omega _{*}^{2}d. \end{aligned} \end{aligned}$$
(6.13)

Considering \(\omega _{*}^{2}<1\), the above formula is impossible for case \(d>0.\) Hence, by (6.10), we get \(I(u)=0\), which means \(u\in N.\) Combining (6.9) and definition of d, we deduce that \(J(u)=d\). \(\square \)

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Di, H., Qiu, Y. Well-posedness, asymptotic stability and blow-up results for a nonlocal singular viscoelastic problem with logarithmic nonlinearity. Z. Angew. Math. Phys. 75, 34 (2024). https://doi.org/10.1007/s00033-023-02177-5

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