Radon measure-valued solutions of first order scalar conservation laws
-
Michiel Bertsch
Abstract
We study nonnegative solutions of the Cauchy problem
where
1 Introduction
In this paper we consider the Cauchy problem
where
Problem (P) with a superlinear φ of the type
When
whose solution is trivially the translated of
It is natural to ask what happens if φ is sublinear. To address this case we must consider solutions of problem (P) which, for
(see Definition 3.3).
Here the measure
Measure-valued entropy solutions are defined similarly (see Definition 3.3).
We use an approximation procedure to construct measure-valued entropy solutions of problem (P) (see Theorem 3.7).
In addition, we prove that the singular part
Concerning the case when φ is sublinear, the following example is particularly instructive:
with
The function in (1.2) is increasing and concave, with
Proposition 1.1.
Let
Let
and
(1.3)Then
Let
If
(1.4)is the unique constructed entropy solution of problem ( 1.1 )–( 1.2 ).
Let us define the waiting time
(by abuse of language, we call
More precisely, if
Since
Proposition 1.2.
Positive waiting times occur in problem (1.1) if and only if the map
The above result is generalized to problem (P), by Theorem 3.18, for functions φ which satisfy for u large a condition implying either concavity or convexity (see assumption (H4) and Remark 3.13).
The proof of Theorem 3.18 makes use of estimates of the density
Another interesting feature of the solution of (1.1)–(1.2), with
Namely, the regular part
Apart from the intrinsic mathematical interest of problem (P), it is worth pointing out its connection with a class of relevant models. Ion etching is a common technique for the fabrication of semiconductor devices, also relevant in other fields of metallurgy, in which the material to be etched is bombarded with an ion beam (see [16, 25, 24]). Mathematical modelling of the process leads to the Hamilton–Jacobi equation in one space dimension
where
In the context of conservation laws, the term “measure-valued solution” usually refers to solutions in the sense of Young measures, after DiPerna’s seminal paper [11]. We stress that this concept of “statistical solutions” is completely different from that of Radon measure-valued solutions, introduced by Demengel and Serre [10], and discussed in the present paper. On the other hand, we do use Young measures in this paper, since they are an important ingredient in the construction of Radon measure valued solutions (see Section 3 and, in particular, Section 5).
A number of ideas used in the present paper go back to papers dealing with Radon measure-valued solutions of quasilinear parabolic problems, also of forward-backward type (in particular, see [6, 4, 5, 21, 23, 27]).
The results presented in this paper naturally lead to some open problems. Among them we mention a general statement about an instantaneous regularizing effect for fluxes with superlinear growth (singular parts should disappear instantaneously for
The paper is organized as follows. In Section 2 we recall several known results used in the sequel and introduce some notation. In Section 3 we present the main results of the paper. In Section 4 we introduce the approximation procedure needed for the construction of solutions. Sections 5–7 are devoted to the proofs of existence, qualitative properties and uniqueness of solutions.
2 Preliminaries
2.1 Function spaces and Radon measures
We denote by
We denote by
The Lebesgue measure, either on
We denote the duality map between
Every
The restriction
We shall use measures
Definition 2.1.
We denote by
if
(2.1)the map
Accordingly, we set
Remark 2.2.
The definition implies that for all
If
for
where
In view of (2.2)–(2.3), we shall always identify the quantities which appear on either side of equalities (2.3).
For any
for any
2.2 Young measures
We recall the following result [2].
Theorem 2.3.
Let
for any open neighborhood U of K in
for every continuous function
where
(2.4)
Suppose further that
for every
given any measurable subset
(2.6)for all continuous functions
Below we shall always refer to the family
Remark 2.4.
Condition (2.5) is very weak. It is equivalent to the statement that for any
Therefore, Theorem 2.3 applies to bounded sequences
If
Theorem 2.5.
Let
where
3 Main results
Throughout the paper we assume that
Hence, there exists
3.1 Definition of solution
In the following definitions, we denote by
the derivative of any
Definition 3.1.
By a solution of problem (P) in the sense of Young measures, we mean a pair
- (3.3)
where
for all
(3.4)where
(3.5)
By an entropy solution of problem (P) in the sense of Young measures, we mean a solution such that
for all ζ as above,
E is convex,
In (3.6), for a.e.
Entropy subsolutions (respectively supersolutions) of problem (P) in the sense of Young measures are defined by requiring that inequality (3.6) be satisfied for all ζ and
Observe that choosing
Remark 3.2.
By (C1), the functions E, F have at most linear growth. Arguing as in (i), it follows that
Definition 3.3.
A measure
where
Entropy subsolutions (respectively supersolutions) of problem (P) are defined by requiring (3.9) to be satisfied for all ζ and
A solution of problem (P) is also a solution in the sense of Young measures.
Moreover, it follows from (3.1) that
Remark 3.4.
If
whence
For the Kružkov entropies
(3.10)
The following proposition states that for any solution of (P) in the sense of Young measures, the map
Proposition 3.5.
Let (H1) be satisfied, let
The map
3.2 Existence and monotonicity
The existence of solutions is proven by an approximation procedure.
If
(e.g., see [23, Lemma 4.1]). Consider the approximating problem
Let us recall the definition of entropy solution of problem (Pn) (e.g., see [9]).
Definition 3.6.
A function
Entropy solutions are weak solutions if
By studying the limiting points of the sequence
Theorem 3.7.
Hypothesis (C1) fails if for example φ is affine in an interval
The following proposition shows that the singular part of an entropy subsolution of (P) does not increase along the lines
Proposition 3.8.
Let (H1) be satisfied.
Let u be an entropy subsolution of problem ( (P) ) in the sense of Young measures. Then
(3.18)In particular,
(3.19)whence
Let u be a solution of problem ( (P) ). Then there is conservation of mass, i.e.,
The linear case
3.3 Waiting time and regularity
It is convenient to distinguish two cases:
3.3.1 Sublinear growth
Beside (H1), we will use the following assumption:
By (H2) the map
Remark 3.9.
The following examples show that all values of
The following property of constructed entropy solutions plays an important role as a uniqueness criterion (see its generalized form given by Proposition 3.17 and Theorem 3.22 below).
Proposition 3.10.
Let (H1)–(H2) be satisfied, and let φ be bounded in
Theorem 3.11.
Let (H1) be satisfied, let
(3.21)Let (H1) – (H2) be satisfied, and let u be the entropy solution of problem ( (P) ) given by Theorem 3.7.
If φ is bounded in
(3.22)If φ is unbounded in
Remark 3.12.
Concerning estimates (3.21) and (3.22), it is worth considering the case in which
Remark 3.13.
In part (ii) of Theorem 3.11, it is enough to require condition (H2) for large values of u. More precisely (see Remark 6.10), Theorem 3.11 (ii) remains valid if instead of (H2), for some
the function
In this connection, observe that the conditions
Let us finally mention the following regularization result.
Proposition 3.14.
Let (H1)–(H2) be satisfied, and let φ be bounded in
Remark 3.15.
It suffices to prove Proposition 3.10, Theorem 3.11 and Proposition 3.14 by assuming
for every
Here
3.3.2 Linear growth
Let φ satisfy the following assumption:
(observe that (H4) reduces to (H2) if
Remark 3.16.
It is easily seen that if u is a solution (respectively an entropy solution) of problem (P), then
for any
By Remark 3.16, the above results for the case
Proposition 3.17.
Let (H1) and (H4) be satisfied, and let
Theorem 3.18.
Let (H1) and (H4) be satisfied, and let u be the entropy solution of problem ( (P) ) given by Theorem 3.7.
Let
Let
Again, Theorem 3.18 (ii) remains valid if, for some
3.4 Uniqueness
In connection with equality (3.11), observe that if
Proposition 3.20.
Let (H1) be satisfied. Let
All entropy solutions u of problem ( (P) ) satisfy
Let us mention that the above statement (ii) holds for any
The following uniqueness result will be proven in Section 7.
Theorem 3.21.
Let (H1) be satisfied and let
By Propositions 3.17, 3.20 and Theorem 3.21, we have the following existence and uniqueness result (observe that (H4) implies C1).
Theorem 3.22.
Let (H1) and (H4) be satisfied, and let
Remark 3.23.
Conditions (3.24)–(3.25) in Theorem 3.22 cannot be omitted. In fact, there exist entropy solutions of problem (P) which do not satisfy either (3.24) or (3.25), depending on φ. Therefore, by Proposition 3.17, they are different from those given by Theorem 3.7, thus uniqueness fails.
For example, let
where
Remark 3.24.
If
and
To prove (3.29) we fix
4 Approximating problems
In this section we consider problem (Pn).
Let
Let
(here
(where
Lemma 4.1.
Let
Lemma 4.2.
Let φ satisfy (3.1).
Then there exists
Proof.
Let
Multiplying the first equation in (4.3) by
Hence, for all
By (3.1) and the definition of the function
Choose
with any
for all
Lemma 4.3.
Let φ satisfy (3.1) and let
Then there exists
If, moreover,
and
Proof.
Inequality (4.13) follows immediately from (4.7) and (4.12).
To prove that
Since
Then it follows from (4.15) that
and, by (4.5) and (4.13), there exists a constant
On the other hand, by (4.5) and since
whence the result follows. ∎
From the above lemmata, we get the following convergence results.
Lemma 4.4.
If
(4.18)(4.19)Moreover,
(4.20)Let φ satisfy ( 3.1 ), let
(4.21)Then
(4.22)
Proof.
By (4.4),
It remains to prove (4.19) and (4.20).
We claim that for a.e.
Set
for some
whence
Since
By (4.2), (4.5), (4.6),
and the Fréchet–Kolmogorov theorem,
and (4.20) follows from (4.5). Finally, (4.19) follows from (4.5), (4.25) and the dominated convergence theorem. ∎
Proposition 4.5.
Let
Moreover, given
Proof.
Let ζ and E be as in Definition 3.6, and
where
Since
The remaining terms in
(4.28) (with
Inequality (4.26) follows from (4.6) and (4.25). Concerning (4.27), it follows from (3.17) that for all
Let
since
and (4.27) follows from (4.29).
Finally, let us show that
and, by (4.16) and the lower semicontinuity of the total variation in
with
5 Existence and monotonicity: Proofs
We proceed with the proof of Theorem 3.7.
Proposition 5.1.
Let (H1) hold and let
For all
where
Proof.
By (4.20), there exist
Since by (4.20) the sequence
Then the result follows by Theorem 2.5 and a standard diagonal procedure. ∎
Remark 5.2.
The function
Proposition 5.3.
Let (H1) hold, let μ be as in (5.4) and let
then, for all
where
Remark 5.4.
If
Moreover,
Proof of Proposition 5.3.
For all
For any
Since
in
Similarly, by (5.1), (5.2), (5.4) and (5.10), with
From (5.9)–(5.12), for any
Since
and
by letting
whence
From the above inequalities, the conclusion follows. ∎
Proposition 5.5.
Let (H1) hold.
Let μ, U and
as
Remark 5.6.
Choosing
If
Proof of Proposition 5.5.
(i) Let us first prove (5.14) for
where
(ii) Next we prove (5.14) for all
where we have used Chebychev’s inequality and the inequality
Letting
Since
Then, letting
(iii) Now let
and, by (5.8), for all
Since
as
Then we obtain that
with
To complete the proof of (5.14), we show that
By (5.21),
for a.e.
Letting
and (5.14) follows from the arbitrariness of ε.
Finally, (5.15) follows from (5.14), the separability of
Proposition 5.7.
Let (H1) hold. Then (5.4) is the Lebesgue decomposition of u, i.e.,
Proof.
Let U be a convex function with
for all
Let
and
as
belong to
By (5.15) and a diagonal argument, there exist a null set
Since
Setting
for all
we have that
Let
The following result is based on the concept of compensated compactness (e.g., see [13]).
Proposition 5.8.
Let (H1) hold.
Then
Proof.
Let
for all
for some
Let
where
for some
where
Let
By (5.36),
Since
in
By a similar argument,
By (5.34) and (5.37),
For every U as above with
Let
By (3.1),
thus
where
since
and Proposition 5.8 follows from the arbitrariness of A.
It remains to prove (5.43) and (5.44). By (5.42) and the monotone convergence theorem,
Since
On the other hand,
Arguing as before, one can show that the first term in the right-hand side of (5.47) vanishes as
for some
To prove the second part of Theorem 3.7, we need the following result which characterizes the disintegration of the Young measure τ.
Proposition 5.9.
Let (H1) hold and
there exist
Then, for a.e.
If
If
(5.49)If
(5.50)where
Proof.
Let
So let
for
By standard approximation arguments, (5.40) is satisfied with
Letting
for all
whence
Similarly, let
Then
Letting
By (C1) and (C3), we can distinguish two cases.
(a) If φ is strictly convex or strictly concave in
where
This implies that
Hence,
Similarly, if φ is strictly convex or strictly concave in
where
It follows that
Remark 5.10.
If (C1) is satisfied for all
Now we can prove Theorem 3.7.
Proof of Theorem 3.7.
Let
(see also (5.25)).
Letting
(in this regard, see also (3.15)).
Thus, the function
Let us end this section by proving Proposition 3.8.
Proof of Proposition 3.8.
For every
where, for a.e.
As in the proof of Proposition 5.7, we have
Let
for some null set
and
Similarly, let
Arguing as in the last part of the proof of Proposition 5.7, we obtain (3.18) and (3.19) from, respectively, (5.56) and (5.57) (we omit the details).
(ii) It follows from (3.8) that for a.e.
where
Since
6 Regularity: Proofs
The first regularity result which we prove is Proposition 3.5. Hence, we need the following lemma.
Lemma 6.1.
Let (H1) be satisfied.
Let
Proof.
Since
The proof of (6.1) is based on (3.4) and (6.3). Let
to get
Letting
Proof of Proposition 3.5.
Let
Similarly, it follows from (6.2) that
To prove (3.13), we observe that, given
Let us now prove the results of Section 3.3.
As explained there, replacing x by
(Recall that in this case
First we prove some estimates of the constructed entropy solutions.
As already said, these estimates are analogous to the Aronson–Bénilan inequality for the convex case
Proposition 6.2.
Let (H1) and (H5) be satisfied, and let
u be an entropy solution of problem (P) given by Theorem 3.7. Then, for a.e.
Moreover, if
there exists
then
Remark 6.3.
If
(see Remark 3.9).
Similarly, if
Let (H5) hold. To prove Proposition 6.2, we use a different regularization of (Pn), that is,
where
From (6.6), for every E convex,
Arguing as in the proof of Proposition 4.5 and letting
So
Lemma 6.4.
Let (H1) and (H5) be satisfied, and let
Lemma 6.5.
Let (H1) and (H5) be satisfied. Then
for all
Proof.
For convenience, we set
It follows from (H5) and a straightforward calculation that
in S. Since
Proof of Proposition 6.2.
Let
for all
for a.e.
Let
Since
for all
Let
whence, by (6.14) and the lower semicontinuity of the total variation,
Similarly, by (5.6), (5.25) and Proposition 5.8,
and, by (6.14) and the lower semicontinuity of the total variation,
It remains to prove that
where we have used (6.13).
We let
By (5.16) and the lower semicontinuity of the total variation,
So we may define
To prove Proposition 3.10, we need the following lemma.
Lemma 6.6.
Let (H1) be satisfied, and let u be the solution of problem (P) given by Theorem 3.7.
Let
Proof.
Let
Setting
Proof of Proposition 3.10.
As pointed out above, it suffices to prove equality (3.20) by assuming (H5).
Let
By the proof of Lemma 6.5, for all
where
(observe that by (H5) we have
Let
Since Ψ is continuous, by Lemma 6.6 and Remark 5.10 (recall that φ satisfies (C1), since φ is strictly concave by assumption H5), letting
whence, by the invertibility of Ψ,
Letting
To prove Theorem 3.11, we need the following result.
Proposition 6.7.
Let (H1) be satisfied. Let
the map
for all
(6.19)(6.20)
Remark 6.8.
It is easily seen that for
where now
Proof of Proposition 6.7.
(i) By (3.1),
Let
Letting
Hence, the distributional derivative
(ii) We set, for
By standard regularization arguments, we can choose
By the dominated convergence theorem,
Hence, (6.19) follows from (6.25). The proof of (6.20) is similar. ∎
Remark 6.9.
Observe that, by (3.18) and (6.21) with
with Φ defined by (6.22).
Now we are ready to prove Theorem 3.11 and Proposition 3.14. As pointed out at the beginning of this section, in doing so it is not restrictive to assume that (H5) holds.
Proof of Theorem 3.11.
(i) By (6.20), for a.e.
whence
(ii) Let
Letting
Let
Letting
Letting
(recall that we have assumed
This proves claim (ii) (a).
If
Remark 6.10.
As we claimed in Remark 3.13, in Theorem 3.11 (ii), we may relax hypothesis (H2) to (H3), with
(
Following the proof of Theorem 3.7, the sequence
Proof of Proposition 3.14.
By the proof of Proposition 3.10, inequality (6.18) is satisfied for a.e.
If
We continue this construction recursively as long as
Hence, this construction stops at some
Since
7 Uniqueness: Proofs
Again, without loss of generality, we may assume that
Proof of Proposition 3.20.
(i) The first step of the proof consists in showing that
Let
Let
where
Hence,
First we let
for a.e.
By (3.15),
Let
It follows from (3.8) and (3.13) that for each
For sufficiently small
Since
If
Letting
whence, by the arbitrariness of σ,
A similar argument shows that
To complete the proof of (3.27), observe that by (3.19) we have
for all
Then (3.27) follows.
(ii) Let
Let u be an entropy solution of problem (P), thus
and, for all
Relying on (7.9)–(7.10) we can conclude the proof by an argument similar to that used in (i). Let
Let
Since
Moreover, by arguing as in (7.6) and (7.7) with
As in the proof of claim (i), combining (7.12) and (7.13) gives
(by a similar argument,
Since
and claim (ii) follows.
Finally, it remains to prove (7.9) (the proof of (7.10) is analogous).
Let
We apply Kružkov’s method of doubling variables. Let
and
whence
We choose
where
Then
Now (7.9) follows by letting
Analogously, it can be proven that, as
In order to prove (7.16), for every sequence
and observe that
Thus, by a variant of the dominated convergence theorem (e.g., see [15, Theorem 4, Section 1.3]), we have
Proof of Theorem 3.21.
Without loss of generality, we may assume that φ is nondecreasing, see Remark 3.15. By Theorem 3.11 (i),
Let us first prove that
To this end, let
and, for all
(recall that φ, by assumption, is increasing).
We must show that (7.18) and (7.19) imply (7.17).
For this purpose, let
with
By the dominated convergence theorem, as
Since
Hence, by (7.20),
and, by a proper choice of h,
(recall that
Next let us prove that
By (3.8) and (7.17), for every
Arguing as in the proof of Lemma 6.1, there exists a null set
If
We can argue as in the proof of (7.17) to obtain that inequality (7.21) holds for every
Let us finally prove Proposition 1.1.
Proof of Proposition 1.1.
A calculation proves that the solution defined by (1.3) if
Remark 7.1.
It is instructive to describe the approximation procedure which gives the solutions mentioned in Proposition 1.1. Consider the family of approximating problems
with φ given by (1.2). It is easily seen that the entropy solution of
for
Hence, for
Letting
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- Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
- Generalized solutions of variational problems and applications
- Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
- Nonlinear Sherman-type inequalities
- Global regularity for systems with p-structure depending on the symmetric gradient
- Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
- Noncoercive resonant (p,2)-equations with concave terms
- Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
- Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
- Localization and multiplicity in the homogenization of nonlinear problems
- Remarks on a nonlinear nonlocal operator in Orlicz spaces
- A Picone identity for variable exponent operators and applications
- On the weakly degenerate Allen-Cahn equation
- Continuity results for parametric nonlinear singular Dirichlet problems
- Construction of type I blowup solutions for a higher order semilinear parabolic equation
- Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
- Comparison results for nonlinear divergence structure elliptic PDE’s
- Constant sign and nodal solutions for parametric (p, 2)-equations
- Monotonicity formulas for coupled elliptic gradient systems with applications
- Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
- A class of semipositone p-Laplacian problems with a critical growth reaction term
- The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
- Reconstruction of Tesla micro-valve using topological sensitivity analysis
- Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
- Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
- Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
- Homoclinics for singular strong force Lagrangian systems
- A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
- On a class of nonlocal nonlinear Schrödinger equations with potential well
- Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
- Regularity for minimizers for functionals of double phase with variable exponents
- Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
- Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
- A-priori bounds for quasilinear problems in critical dimension
- Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
- On the Sobolev space of functions with derivative of logarithmic order
- On a logarithmic Hartree equation
- Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
- Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
- Existence for (p, q) critical systems in the Heisenberg group
- Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
- Some hemivariational inequalities in the Euclidean space
- Existence of standing waves for quasi-linear Schrödinger equations on Tn
- Periodic solutions for second order differential equations with indefinite singularities
- On the Hölder continuity for a class of vectorial problems
- Bifurcations of nontrivial solutions of a cubic Helmholtz system
- On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
- Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2
- Positive solutions for diffusive Logistic equation with refuge
- Null controllability for a degenerate population model in divergence form via Carleman estimates
- Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
- On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients
- Multiplicity and concentration results for magnetic relativistic Schrödinger equations
- Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
- The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
- Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
- Pseudo almost periodic solutions for a class of differential equation with delays depending on state
- Normalized multi-bump solutions for saturable Schrödinger equations
- Some inequalities and superposition operator in the space of regulated functions
- Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
- Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
- Morrey estimates for a class of elliptic equations with drift term
- A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
- Global and non global solutions for a class of coupled parabolic systems
- On the analysis of a geometrically selective turbulence model
- Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
- Lack of smoothing for bounded solutions of a semilinear parabolic equation
- Gradient estimates for the fundamental solution of Lévy type operator
- π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
- On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
- Anisotropic problems with unbalanced growth
- On a fractional thin film equation
- Minimum action solutions of nonhomogeneous Schrödinger equations
- Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
- Optimal rearrangement problem and normalized obstacle problem in the fractional setting
- A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
Articles in the same Issue
- Frontmatter
- On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
- Regularity of solutions of the parabolic normalized p-Laplace equation
- Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
- Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
- Radon measure-valued solutions of first order scalar conservation laws
- Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
- Generalized solutions of variational problems and applications
- Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
- Nonlinear Sherman-type inequalities
- Global regularity for systems with p-structure depending on the symmetric gradient
- Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
- Noncoercive resonant (p,2)-equations with concave terms
- Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
- Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
- Localization and multiplicity in the homogenization of nonlinear problems
- Remarks on a nonlinear nonlocal operator in Orlicz spaces
- A Picone identity for variable exponent operators and applications
- On the weakly degenerate Allen-Cahn equation
- Continuity results for parametric nonlinear singular Dirichlet problems
- Construction of type I blowup solutions for a higher order semilinear parabolic equation
- Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
- Comparison results for nonlinear divergence structure elliptic PDE’s
- Constant sign and nodal solutions for parametric (p, 2)-equations
- Monotonicity formulas for coupled elliptic gradient systems with applications
- Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
- A class of semipositone p-Laplacian problems with a critical growth reaction term
- The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
- Reconstruction of Tesla micro-valve using topological sensitivity analysis
- Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
- Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
- Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
- Homoclinics for singular strong force Lagrangian systems
- A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
- On a class of nonlocal nonlinear Schrödinger equations with potential well
- Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
- Regularity for minimizers for functionals of double phase with variable exponents
- Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
- Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
- A-priori bounds for quasilinear problems in critical dimension
- Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
- On the Sobolev space of functions with derivative of logarithmic order
- On a logarithmic Hartree equation
- Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
- Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
- Existence for (p, q) critical systems in the Heisenberg group
- Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
- Some hemivariational inequalities in the Euclidean space
- Existence of standing waves for quasi-linear Schrödinger equations on Tn
- Periodic solutions for second order differential equations with indefinite singularities
- On the Hölder continuity for a class of vectorial problems
- Bifurcations of nontrivial solutions of a cubic Helmholtz system
- On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
- Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2
- Positive solutions for diffusive Logistic equation with refuge
- Null controllability for a degenerate population model in divergence form via Carleman estimates
- Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
- On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients
- Multiplicity and concentration results for magnetic relativistic Schrödinger equations
- Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
- The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
- Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
- Pseudo almost periodic solutions for a class of differential equation with delays depending on state
- Normalized multi-bump solutions for saturable Schrödinger equations
- Some inequalities and superposition operator in the space of regulated functions
- Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
- Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
- Morrey estimates for a class of elliptic equations with drift term
- A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
- Global and non global solutions for a class of coupled parabolic systems
- On the analysis of a geometrically selective turbulence model
- Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
- Lack of smoothing for bounded solutions of a semilinear parabolic equation
- Gradient estimates for the fundamental solution of Lévy type operator
- π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
- On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
- Anisotropic problems with unbalanced growth
- On a fractional thin film equation
- Minimum action solutions of nonhomogeneous Schrödinger equations
- Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
- Optimal rearrangement problem and normalized obstacle problem in the fractional setting
- A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
