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Stabilization of Quasilinear Parabolic Equations by Cubic Feedback at Boundary with Estimated Region of Attraction

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Pith's one-line read Cubic boundary feedback can exponentially stabilize quasilinear parabolic PDEs that blow up in open loop, with a computable region of attraction, full classical well-posedness, and positivity preservation.

desk verdict A genuinely new Lyapunov framework for 1-D quasilinear blow-up PDEs with cubic boundary feedback, but Theorem 1 has a real hypothesis gap for u^2 u_x and the inverse formula is wrong. read the letter →

arxiv 2506.18634 v1 pith:OVDZDRWN submitted 2025-06-23 math.AP

Mohamed Camil Belhadjoudja (GIPSA-INFINITY) , M Maghenem (GIPSA-INFINITY) , E Witrant (GIPSA-INFINITY) , M Krstic (UC San Diego) This is my paper · ORCID
classification math.AP MSC 35K5535B4493D1593C20
keywords quasilinearparabolicPDEsfinite-timeblow-upboundarystabilizationcubicfeedbackregionofattractionLyapunovmethodswell-posednesspositivitypreservation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quasilinear parabolic equations of the form $u_t=\varepsilon(u)u_{xx}+\sum_{i=1}^n \gamma_i u^i u_x+u^p$ on the unit interval, with zero boundary inputs, can blow up in finite time even for arbitrarily small initial data. This paper constructs boundary feedback laws that are cubic polynomials of the boundary values, $u_x(0)=\lambda_0 u(0)+\mu u(0)^3$ and $u_x(1)=-\lambda_1 u(1)-\mu u(1)^3$, and proves that, under three explicit algebraic inequalities on the gains and system parameters, the zero state is exponentially stable in $L^2$ and $H^1$ with a region of attraction estimated by $\kappa_0(u_0)\le\sqrt{2\omega}$. The same theorem establishes existence and uniqueness of complete classical solutions, convergence to zero in $H^2$ and $C^1$, and nonnegativity of solutions starting from nonnegative data. Because the controllers use only boundary measurements, the design covers state-dependent diffusion, convection, and destabilizing superlinear reaction, a regime where existing backstepping boundary controls could not be applied. This is the natural form of the result for this class, since global stabilization is known to be impossible in general.

What carries the argument

The carrying object is the Lyapunov functional $E=V+H$, where $V=\frac12|u|_{L^2}^2$ and $H=\frac12|u_x|_{L^2}^2+\frac{\lambda_1}{2}u(1)^2+\frac{\mu}{4}u(1)^4+\frac{\lambda_0}{2}u(0)^2+\frac{\mu}{4}u(0)^4$. Along solutions, the paper derives $\dot E\le -\alpha(E)V-\Lambda(E)|u_x|_{L^2}^2-\varepsilon(u(1))\Gamma(E)u(1)^4-\varepsilon(u(0))\Gamma(E)u(0)^4$, with the coefficient functions $\alpha,\Lambda,\Gamma$ made positive by conditions (12)-(14). Agmon's inequality $|u|_\infty^2\le |u|_{L^2}^2+2|u|_{L^2}|u_x|_{L^2}$ converts the sup norm into a function of $E$, turning the reaction- and convection-dependent coefficients into negative dissipation terms. A cutoff version of the system is used to access a classical quasilinear well-posedness theorem, and a comparison principle gives positivity.

What would settle it

A concrete check: in the constant-diffusion case $\varepsilon(u)\equiv\varepsilon$, $\gamma_i=0$, $p=2$, choose $u_0$ with $\kappa_0(u_0)\le\sqrt{2\omega}$ and gains satisfying (12)-(14), and simulate the closed loop; if the sup norm ever exceeds $\sqrt{6\omega}$, or if $|u|_{L^2}$ does not lie below $\|u_0\|_{L^2}e^{-\sigma t}$ with $\sigma$ from (15), then the region-of-attraction estimate or the exponential decay claim fails.

Watch

Extended reading notes

Core claim

Theorem 1 asserts that, for system (1) under Assumption 1, the cubic Neumann boundary laws $u_x(0)=\lambda_0u(0)+\mu u(0)^3$ and $u_x(1)=-\lambda_1u(1)-\mu u(1)^3$ make zero exponentially stable in $L^2$ and $H^1$ with region of attraction $\kappa_0(u_0)\le\sqrt{2\omega}$, provided $\omega$ satisfies the explicit inequalities (12)-(14) and the gains $\lambda_0,\lambda_1,\mu$ are selected as in (10)-(11). Every complete classical solution then has its $H^2$ and $C^1$ norms converge to zero, compatibility conditions give existence and uniqueness of a complete classical solution, and nonnegative initial data give nonnegative solutions. The theorem covers the quasilinear structure itself: the diffusion coefficient may depend on the state, the convection may be polynomial, and the reaction $u^p$ with $p>1$ is the destabilizing term responsible for open-loop blow-up. In many cases the guaranteed region of attraction and the decay rate grow unboundedly as the diffusion increases, so large diffusion acts as a resource for the boundary controller.

Load-bearing premise

All of the stability estimates pass through the one-dimensional Agmon inequality $|u|_\infty^2\le |u|_{L^2}^2+2|u|_{L^2}|u_x|_{L^2}$; this bound has no analogue in higher-dimensional domains, so the Lyapunov argument closes only on $[0,1]$ and the advertised results do not extend to $d>1$ without a new idea.

Editorial extensions

If this is right

  • For the quasilinear class (1), the cubic boundary laws (10)-(11) certify a region of attraction $\kappa_0(u_0)\le\sqrt{2\omega}$ whenever the algebraic checks (12)-(14) hold, so designers can verify stabilizability without solving the PDE.
  • Inside this region, $L^2$ and $H^1$ norms decay exponentially and $H^2$ and $C^1$ norms converge to zero, so the controller delivers both transient stabilization and regularity recovery.
  • The same boundary laws admit Neumann, Dirichlet, and mixed-type implementations via the explicit inverse (17), making the design adaptable to actuation constraints in thermal or fluid systems.
  • Under a parity condition on the convection coefficients, one-sided boundary feedback alone (only at $x=0$ or only at $x=1$) achieves the same three guarantees, reducing actuator requirements.
  • Nonnegative initial data produce nonnegative solutions, which is essential in concentration or temperature models where negative states are unphysical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper leaves implicit: in the constant-diffusion, purely reactive case ($\varepsilon$ constant, $\gamma_i=0$, $p=2$), the guaranteed basin radius should scale with $\varepsilon$; numerical continuation on the one-dimensional equation could map the true basin and compare its asymptotics with the estimate.
  • Because Agmon's inequality is the only 1-D input, a multi-dimensional version would have to replace pointwise sup-norm control with boundary-trace estimates; the paper's conclusion flags exactly this obstruction, so the natural next conjecture is that the basin estimates survive with domain-dependent constants in two or three dimensions.
  • The controller uses only instantaneous boundary values, so it is compatible with observer-based or output-feedback architectures; the paper does not discuss estimation, but the boundary-measurement structure is presumably sufficient.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies a one-dimensional quasilinear parabolic equation (1) with state-dependent diffusion, polynomial convection, and a superlinear reaction term, and proposes cubic Neumann feedback laws (10)-(11). Theorem 1 claims exponential L2 and H1 stability of the origin with an explicit estimate of the region of attraction, exponential H2 and C1 convergence to zero, existence and uniqueness of complete classical solutions, positivity for nonnegative initial data, and invertibility of the boundary maps. The proof combines two Lyapunov functionals V and H, the one-dimensional Agmon inequality, a cutoff argument for well-posedness, and a parabolic comparison principle. The central Lyapunov chain in Lemmas 1-2 and the propagation argument for E(t)≤E(0) are coherent, and the well-posedness route via a truncated system is standard. However, a key hypothesis, condition (14), is unsatisfiable for the quadratic-convection case n=2, γ2≠0, and the explicit inverse formula (17) in Property 4 appears algebraically incorrect.

Significance. If the advertised results were fully valid, this would be a valuable contribution to boundary stabilization of quasilinear parabolic equations with blow-up, since the existing literature is largely limited to semilinear or special Burgers-type problems. The paper's strengths are its explicit and self-contained estimates, the clean two-functional Lyapunov construction, the standard cutoff/regularization route to well-posedness, and the use of the comparison principle for positivity. The paper does not fit data or rely on empirical predictions; all constants are explicit design parameters, which is a positive feature. The authors correctly acknowledge that the one-dimensional Agmon inequality is essential and that higher-dimensional extensions are open. Nevertheless, the advertised scope is currently too broad: condition (14) excludes the natural quadratic-convection subclass, and the explicit Cardano inversion formula contains a clear algebraic error. These issues are fixable but are load-bearing for the theorem as stated.

major comments (2)
  1. [§3, Theorem 1, condition (14)] When n=2 and γ2≠0, which is exactly the quadratic-convection term u^2 u_x motivated in the Introduction through Burgers' equation, condition (14) reduces to |γ2|/4 > |γ2|/4, which is impossible for any choice of M, ω, or the free gains. Consequently, Theorem 1 gives no stabilization guarantee for this advertised subclass. In the proof, the corresponding term Γ(E) in (22) is identically zero for i=2, so the quartic boundary dissipation that the argument relies on is absent. The theorem should either be restricted explicitly to cases where at least one γ_i with i≥3 is nonzero, or a separate case covering i=2 must be proved, e.g., by observing that Γ(E)=0 still leaves a non-positive right-hand side in (22) as long as α(E) and Λ(E) are positive. As written, the mismatch between the advertised class and the hypotheses is a load-bearing gap.
  2. [§4.4, Property 4 and Eq. (17)] The stated Cardano inversion formula does not solve the cubic equation (42). Setting y=u_x(l) and z=u(l), equation (42) is μ z^3 + λ_l z + (-1)^{l+1} y = 0, so the standard solution contains y/(2μ), not y/(2λ_l), and the sign must be adjusted according to l. Thus the formula in (17) is algebraically incorrect and does not establish Property 4 as stated. This is a local error, but it affects the claimed implementation of the controllers in Dirichlet or mixed form; the formula should be corrected and re-verified.
minor comments (4)
  1. [§4.2, after Eq. (39)] The sentence 'u≡0 is the unique complete complete solution to Σ̄' contains a duplicated word; please correct it.
  2. [§3, Eq. (16)] The quantity ζ is used in the displayed formula for ζ̄ before it is defined later in the proof; please define ζ before Eq. (16) or restructure the notation so that the expression is self-contained.
  3. [§5, Conclusion] The conclusion correctly states that the approach does not extend to higher dimensions because Agmon's inequality is one-dimensional. It would be useful to place this limitation earlier, e.g., in the introduction or in the statement of Theorem 1, so that readers do not over-interpret the advertised scope.
  4. [Throughout] The manuscript contains several typographical spacing issues, such as 'L2 andH 1' and 'H ö lder', and author affiliation 'Departement' should be 'Département'. These should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Lyapunov derivation is self-contained and the self-citations are contextual only.

full rationale

The paper is a conditional Lyapunov analysis, not a fit-and-predict scheme: every gain (m0, m1, k0, k1, M) and the radius ω are free design parameters subject to explicit sufficient inequalities (12)-(14), and no parameter is calibrated to solution data and then renamed a prediction. The advertised stability and convergence statements are obtained from the differential inequalities (18), (22), and (30)-(36) under the standing assumption κ0(u0) ≤ √2ω, so the region-of-attraction estimate is an initial-data hypothesis verified by the Lyapunov argument rather than an output smuggled into the hypotheses. Well-posedness is obtained through a cutoff system whose existence and uniqueness are imported from the external theorem [18, Theorem 7.4], and positivity uses the external comparison principle [25]; the self-citations to the authors' prior work [2,3,4] and to [15] are motivational or supply standard, parameter-free inequalities (Agmon and Poincaré) that do not carry the central claim. No self-citation chain is load-bearing, no uniqueness theorem from the authors' own prior work is invoked as an external fact, and no 'prediction' reduces by construction to an input. The skeptic's observation that condition (14) is unsatisfiable when the only higher-order convection is i=2 is a mathematical correctness concern about the sufficient condition, not a circularity of the derivation chain.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No new physical entities are postulated. The main free objects are controller gains and region-radius parameters; the cutoff system Σbar is a proof device, not an invented physical entity. The central result is a parameter-free Lyapunov derivation given the class assumptions and gain choices.

free parameters (4)
  • m0, m1
    Positive Lyapunov gains chosen to satisfy conditions (12)-(13); they set λ0, λ1 and the decay rate σ.
  • k0, k1
    Nonnegative extra feedback gains, free; they only improve the decay estimates.
  • M
    Positive gain appearing in μ and in condition (14); must be large enough for the convection-compensation and cutoff arguments.
  • ω
    Radius parameter of the region-of-attraction estimate; chosen to satisfy (12)-(14). Not fitted to data.
assumptions (6)
  • domain assumption Assumption 1: ε(u) is differentiable, inf ε ≥ ε > 0, and |ε'(u)| ≤ arε(s) for |u| ≤ s with arε continuous and nondecreasing.
    Gives the diffusion lower bound and controls ε(u), ε'(u) in Lemmas 1 and 2.
  • domain assumption u0 ∈ H^{2+β}[0,1] and the 0th-order compatibility condition (4) hold.
    Needed for the existence of classical solutions, as stated in Definition 1 and Remark 1.
  • standard math Ladyzhenskaya-Solonnikov-Ural'tseva Theorem 7.4 (Lemma 6) gives a unique complete solution to the cutoff system Σbar under structure conditions.
    Existence and uniqueness for Σbar is imported from the classical quasilinear parabolic theory.
  • standard math Pao's parabolic comparison principle (Lemma 7) applies to the positive orthant.
    Used to prove Property 3; requires monotone reaction and boundary maps on [0,+∞) and Hölder coefficients.
  • standard math Agmon's and Poincaré's inequalities on [0,1] (Lemmas 4 and 5).
    Agmon bounds |u|∞ by E; Poincaré moves boundary terms into V and |ux|² terms.
  • domain assumption For Property 2, ε ∈ C²; for Property 3, ε' ∈ H_loc(R).
    Extra smoothness hypotheses stated in Theorem 1 for the well-posedness and positivity conclusions.

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Pith. "Pith review of Stabilization of Quasilinear Parabolic Equations by Cubic Feedback at Boundary with Estimated Region of Attraction." pith.science (2026). https://pith.science/paper/OVDZDRWN

@misc{pith2026250618634,
  author       = {Pith},
  title        = {Pith review of: Stabilization of Quasilinear Parabolic Equations by Cubic Feedback at Boundary with Estimated Region of Attraction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVDZDRWN}},
  note         = {Machine review of arXiv:2506.18634}
}
read the original abstract

For quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow up in open loop, i.e., under null boundary conditions, we provide an estimate of the region of attraction under cubic feedback laws applied at the boundary, using boundary measurements. We guarantee: 1-L 2 and H 1 exponential stability of the origin with an estimate of the region of attraction. 2-Convergence of the H 2 and the C 1 norms of the solutions to zero. 3-Existence and uniqueness of complete classical solutions. 4-Positivity of the solutions starting from positive initial conditions. Unlike existing approaches, our framework handles nonlinear state-dependent diffusion, convection, and (destabilizing) reaction. The cubic terms are used to enlarge our estimate of the region of attraction. The size of the region of attraction is shown, in many cases, to grow unboundedly as diffusion increases. Finally, our controllers can be implemented as Neumann, Dirichlet, or mixed-type boundary conditions.

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Works this paper leans on

34 extracted references · 33 canonical work pages

  1. [1]

    Balogh and M

    A. Balogh and M. Krstic, Burgers’ equation with nonlinear boundary feedback: H 1 stability, well posedness, and simulation, Mathematical Problems in Engineering, 6(2-3), 189-200, 2000

    work page 2000

  2. [2]

    M. C. Belhadjoudja, M. Krstic, M. Maghenem, and E. Witrant, From Sontag’s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization, In Proceedings of the 2024 IEEE American Control Conference (ACC), pp. 3296-3301

    work page 2024

  3. [3]

    M. C. Belhadjoudja, M. Maghenem, E. Witrant, and M. Krstic, Boundary Stabilization of Quasilinear Parabolic PDEs that Blow Up in Open Loop for Arbitrarily Small Initial Conditions, Submitted to the 2025 IEEE Conference on Decision and Control. HAL Preprint: hal-05069716

    work page 2025

  4. [4]

    M. C. Belhadjoudja, M. Maghenem, E. Witrant, and C. Prieur, Adaptive Boundary Control of the Kuramoto- Sivashinsky Equation Under Intermittent Sensing, Accepted for Publication in Automatica, 2025

    work page 2025

  5. [5]

    P. D. Christofides, Robust control of parabolic PDE systems, Chemical Engineering Science, 53(16), 2949-2965, 1998

    work page 1998

  6. [6]

    J. M. Coron and E. Tr´ elat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM journal on control and optimization, 43(2), 549-569, 2004

    work page 2004

  7. [7]

    Coussy, Poromechanics, John Wiley & Sons, 2004

    O. Coussy, Poromechanics, John Wiley & Sons, 2004

    work page 2004

  8. [8]

    Fern´ andez-Cara, and E

    E. Fern´ andez-Cara, and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire. Vol. 17. No. 5, 2000

    work page 2000

Show all 34 references
  1. [9]

    Fujita, On the blowing up of solutions to the Cauchy problem for ut = ∆u + u1+α, J

    H. Fujita, On the blowing up of solutions to the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 13, 109–124, 1966

    1966

  2. [10]

    V. A. Galaktionov, and J. L. V´ azquez, The problem of blow- up in nonlinear parabolic equations, Discrete and continuous dynamical systems 8.2: 399-434, 2002

    2002

  3. [11]

    Karafyllis and M

    I. Karafyllis and M. Krstic, Input-to-state stability for PDEs, Springer International Publishing, 2019

    2019

  4. [12]

    Karafyllis and M

    I. Karafyllis and M. Krstic, Global stabilization of a class of nonlinear reaction-diffusion partial differential equations by boundary feedback, SIAM Journal on Control and Optimization, 57(6), 3723-3748, 2019

    2019

  5. [13]

    Karafyllis and M

    I. Karafyllis and M. Krstic, Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of One- Dimensional Semilinear Parabolic PDEs, SIAM Journal on Control and Optimization, 57(3), 016-2036, 2019

    2019

  6. [14]

    Karafyllis, Lyapunov-based boundary feedback design for parabolic PDEs, International Journal of Control, 94(5), 1247-1260, 2019

    I. Karafyllis, Lyapunov-based boundary feedback design for parabolic PDEs, International Journal of Control, 94(5), 1247-1260, 2019

    2019

  7. [15]

    Krstic, On global stabilization of Burgers’ equation by boundary control, Systems & Control Letters, 37(3), 123-141, 1999

    M. Krstic, On global stabilization of Burgers’ equation by boundary control, Systems & Control Letters, 37(3), 123-141, 1999

    1999

  8. [16]

    Krstic, L

    M. Krstic, L. Magnis and R. Vazquez, Nonlinear Stabilization of Shock-Like Unstable Equilibria in the Viscous Burgers PDE, in IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 1678-1683, 2008 . 8

    2008

  9. [17]

    A. A. Lacey, Diffusion models with blow-up, Journal of computational and applied mathematics 97.1-2: 39-49, 1998

    1998

  10. [18]

    O. A. Ladyzhenskaia, V. A. Solonnikov, and N. N. Ural’tseva, Linear and quasi-linear equations of parabolic type, (Vol. 23). American Mathematical Soc., 1968

    1968

  11. [19]

    Lhachemi and C

    H. Lhachemi and C. Prieur, Finite-dimensional observer- based boundary stabilization of reaction–diffusion equations with either a Dirichlet or Neumann boundary measurement, Automatica, 135:109955, 2022

    2022

  12. [20]

    Liu and M

    W-J. Liu and M. Krstic, Adaptive control of Burgers’ equation with unknown viscosity, International Journal of Adaptive Control and Signal Processing, 15(7), 745-766, 2001

    2001

  13. [21]

    Liu and M

    W-J. Liu and M. Krstic, Global boundary stabilization of the Korteweg-de Vries-Burgers equation, Computational and Applied Mathematics, 21(1), 315-354, 2002

    2002

  14. [22]

    Liu and M

    W-J. Liu and M. Krstic, Stability enhancement by boundary control in the Kuramoto–Sivashinsky equation, Nonlinear Analysis: Theo., Meth. & Appli., 43(4), 485-507, 2001

    2001

  15. [23]

    Maghenem, C

    M. Maghenem, C. Prieur, and E. Witrant, Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent Data Availability, In Proceedings of the 2022 IEEE American Control Conference (ACC), pp. 2227-2232

    2022

  16. [24]

    Mazenc and C

    F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Mathematical Control and Related Fields, 1(2):231–250, 2011

    2011

  17. [25]

    Pao, Positive solutions of a nonlinear boundary-value problem of parabolic type, J

    C.V. Pao, Positive solutions of a nonlinear boundary-value problem of parabolic type, J. of Diff. Equ., 22, 145-163, 1976

    1976

  18. [26]

    A. A. Samarskii, and A. P. Mikhailov, Blow-up in quasilinear parabolic equations, (Vol. 19). Walter de Gruyter, 2011

    2011

  19. [27]

    Schneider, T

    R. Schneider, T. Rothaug, and P. Benner, Flow stabilisation by Dirichlet boundary control, In PAMM, Vol. 8, No. 1, pp. 10961-10962, 2008

    2008

  20. [28]

    Tarmy, Reactor technology, Kirk-Othmer Encyclopedia of Chemical Technology, 2000

    B.L. Tarmy, Reactor technology, Kirk-Othmer Encyclopedia of Chemical Technology, 2000

    2000

  21. [29]

    Vazquez, J

    R. Vazquez, J. Auriol, F. Bribiesca-Argomedo, and M. Krstic, Backstepping for Partial Differential Equations, arXiv preprint arXiv:2410.15146, 2024

    2024 arXiv

  22. [30]

    Vazquez, and M

    R. Vazquez, and M. Krstic, Control of 1-D parabolic PDEs with Volterra nonlinearities, part I: Design, Automatica, 44(11), 2778-2790, 2008

    2008

  23. [31]

    Vazquez, and M

    R. Vazquez, and M. Krstic, Control of 1-D parabolic PDEs with Volterra nonlinearities, part II: Analysis, Automatica, 44(11), 2791-2803, 2008

    2008

  24. [32]

    Zekraoui, N

    S. Zekraoui, N. Espitia, and W. Perruquetti, Lyapunov- based nonlinear boundary control design with predefined convergence for a class of 1D linear reaction-diffusion equations, European Journal of Control, 74, 100845, 2023

    2023

  25. [33]

    N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhailov, A. A. Samarskii, Localization of thermonuclear combustion in plasma with electron heat conductance, Pis’ma v Zh. Eksper. Teor. Fiz. 26(9) (1977), 620–624; JETP Letters 26 (1977)

    1977

  26. [34]

    Ya. B. Zel’dovich, G. I. Barenblatt, V. B. Librovich, G. M. Makhviladze, The mathematical theory of combustion and explosions, Consultants Bureau, New York, 1985 . A Useful results In this section, we recall some results that are used in the paper. The following two inequaliti...

    1985

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